Master Dissertation
Department of Mathematics
Robin Vandaele
Academic year 2015-2016
Reverse mathematics of the
Browder-Göhde-Kirk fixed point
theorem
Advisors:
Dr. P. Shafer
Prof. Dr. A. Weiermann
Master dissertation submitted in order to obtain the academic degree
of Master of Science in Mathematics, specialization in Pure Mathematics.
Master Dissertation
Department of Mathematics
Robin Vandaele
Academic year 2015-2016
Reverse mathematics of the
Browder-Göhde-Kirk fixed point
theorem
Advisors:
Dr. P. Shafer
Prof. Dr. A. Weiermann
Master dissertation submitted in order to obtain the academic degree
of Master of Science in Mathematics, specialization in Pure Mathematics.
Contents
Preface
i
Permission for use of content
ii
Toelating tot bruikleen
ii
Resume
iii
1 The language of second-order arithmetic
1.1 The formal system Z2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Subsystems of Z2
2.1 RCA0 . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 The number system and primitive recursion
2.1.2 Concepts of analysis and topology . . . . . .
2.2 WKL0 . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 ACA0 . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 ATR0 . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Π11 -CA0 . . . . . . . . . . . . . . . . . . . . . . . .
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3 Reverse mathematics
3.1 The Main Question . . . . . . . . . . . . . . . . . . . . . .
3.2 Brouwer’s fixed point theorem for the unit square . . . . .
3.2.1 The unit square in R2 . . . . . . . . . . . . . . . . .
3.2.2 Proving that Brouwer’s fixed point theorem for the
equivalent to WKL0 over RCA0 . . . . . . . . . . . .
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1
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unit square is
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4 Browder-Göhde-Kirk’s fixed point theorem
4.1 The statement in Z2 . . . . . . . . . . . . . . . . . . . . . . . . . .
b a Hilbert space and K a closed ball
4.2 The reverse mathematics for E
4.2.1 Proving that ACA0 → BGKH over RCA0 . . . . . . . . . . .
4.2.2 The Hilbert space l2 . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Proving that BGKH → ACA0 over RCA0 . . . . . . . . . . .
b a uniformly convex Banach Space .
4.3 The reverse mathematics for E
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61
A Appendix: Nederlandstalige samenvatting
69
B Appendix: Coding continuous functions
B.1 Coding the metric projection on a closed ball in a Hilbert space. . . . . . .
B.2 Coding the metric projection on a half-space in a Hilbert space . . . . . . .
B.3 Coding of a function series in a separable Banach space . . . . . . . . . . .
70
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References
83
Preface
‘God created the natural numbers and mankind did the rest.’ This was one of the first
sentences I heard while studying mathematics at Ghent University. It was only until a few
years later, by studying mathematical logic, I learned that this is not necessarily the case.1
And even so, merely the existence of the natural numbers itself isn’t enough to construct
our perception of modern mathematics. We had to apply rules which seemed plausible
enough, such that we didn’t really have to think about their validity. For example, given
sets A and B, we can define the cartesian product A × B. In ordinary mathematics, this
is something we can do without the need of any further justification.
That’s why I’m so intrigued with the subject of mathematical logic. I can’t seem to
find any other branch of mathematics in which our reasoning is justified more than in
logics. Here, one learns that these ‘rules’ we applied, are actually or follow from some
basic axioms which we can formalize in axiomatic systems. For example ZF or ZFC, in
which we humans ourselves constructed some perception of the natural numbers starting
from some basic axioms. So essentially, one could say God created an axiomatic system
and mankind did the rest. Actually, He created a few. He had a lot of spare time in his
days.
Learning that there is actually a mathematical subject that studies something that
resembles the opposite of our construction of modern mathematics from given axioms,
namely given some ordinary mathematical theorem, looking to which axioms are essentially
needed to formalize and prove this theorem, I knew I had to study this subject of ‘reverse
mathematics’ further. Since my second favourite branch of mathematics is plain analysis,
doing my thesis on a subject which involves reverse mathematics as well as analysis, seemed
to be a perfect idea.
My thanks go to Dr. P. Shafer and Prof. Dr. A. Weiermann for intriguing me with
subjects as mathematical logic and proof theory, and for helping me to find a topic in
which I could fully study the subject of reverse mathematics. Also, Dr. P. Shafer for all
the effort he put the past year into helping me explore this branch of mathematics, which
I essentially knew nothing about in advance, and for coming up with a lot of crucial ideas
and strategies for this thesis.
To end with, I can conclude that my skills in mathematical and logical reasoning
have increased a lot since high school by studying mathematics at Ghent University. The
opposite holds for my skills in solving ordinary integrals.
Robin Vandaele,
May 2016
1
Note that we are not emphasizing the question whether or not there exists a God here. This thesis
does not concern the provability of this statement.
i
Permission for use of content
“The author gives permission to make this master dissertation available for consultation
and to copy parts of this master dissertation for personal use. In the case of any other
use, the limitations of the copyright have to be respected, in particular with regard to the
obligation to state expressly the source when quoting results from this master dissertation.”
Toelating tot bruikleen
“De auteur geeft de toelating deze masterproef voor consultatie beschikbaar te stellen
en delen van de masterproef te kopiëren voor persoonlijk gebruik. Elk ander gebruik
valt onder de beperkingen van het auteursrecht, in het bijzonder met betrekking tot de
verplichting de bron uitdrukkelijk te vermelden bij het aanhalen van resultaten uit deze
masterproef.”
Robin Vandaele,
May 31st, 2016
ii
Resume
In [1], Simpson gives an analysis of Brouwer’s fixed point theorem in his chapter on weak
König’s lemma, proving that Brouwer’s theorem is equivalent to weak König’s lemma over
RCA0 . We shall take a look at another fixed point theorem, namely the Browder-GöhdeKirk fixed point theorem. The purpose of this thesis is to perform and analyze the reverse
mathematics of this theorem.
To make this thesis somewhat self-contained, we shall start with defining the language
of second-order arithmetic L2 and the formal system Z2 in Chapter 1, continuing with the
natural subsystems RCA0 , WKL0 , ACA0 , ATR0 and Π11 -CA0 of Z2 in Chapter 2. RCA0
will be the weakest of these subsystems, i.e., is included in all other subsystems, and we
shall summarize the basic concepts of real analysis and topology that can be developed in
this system, and hence in all the other systems. Chapters 1 and 2 will be largely based on,
or containing notes from [1] and [2].
The purpose of Chapter 3 is to introduce reverse mathematics. The Main Question
shall be: ‘Given a theorem ϕ of ordinary mathematics, what is the weakest subsystem of
Z2 in which ϕ is provable? ’ As an example, we shall prove that Brouwer’s fixed point
theorem for the unit square is equivalent to WKL0 over RCA0 . This proof will be largely
based on Simpson’s proof of the general version of Brouwer’s fixed point theorem in [1]
Section IV.7.
In Chapter 4, we shall take a look at the other fixed point theorem, the BrowderGöhde-Kirk fixed point theorem. We will proof that this theorem, also denoted by BGK,
is equivalent to ACA0 over RCA0 . The direction ACA0 → BGK will be mainly based on
a generalization of an ordinary mathematical proof from [3], by investigating how these
strategies work out within the context of reverse mathematics. The reversal BGK→ ACA0
will be mainly based on ideas from [4].
iii
1
The language of second-order arithmetic
In this chapter we shall define the two-sorted language L2 and the formal system Z2 . By
two-sorted we mean that there are two distinct sorts of variables which are intended to
range over two different kinds of objects:
• Variables of the first sort are known as number variables, are denoted by i, m, n, . . . ,
and are intended to range over the set ω = {0, 1, 2, . . .} of all natural numbers.
• Variables of the second sort are known as set variables, are denoted by X, Y, Z, . . . ,
and are intended to range over all subsets of ω.
Numerical terms are number variables, the constant symbols 0 and 1, and n + m
and n · m whenever n and m are numerical terms. Here + and · are binary operation
symbols in our language. Atomic formulas are n = m, n < m and n ∈ X, where n and
m are numerical terms and X is any set variable. Here =, < and ∈ are binary relation
symbols in our language. Formulas are built up from atomic formulas by means of the
propositional connectives ∧, ∨, ¬, →, ↔, number quantifiers ∀n, ∃n, and set quantifiers
∀X, ∃X. A sentence is a formula with no free variables.
Definition 1.1. (language of second-order arithmetic). L2 is defined to be the language
of second-order arithmetic as described above.
Remark 1.2. As customary, we shall use some obvious abbreviations. For instance
/ X stands for
2 + 2 = 4 stands for (1 + 1) + (1 + 1) = ((1 + 1) + 1) + 1, (m + n)2 ∈
¬((m + n) · (m + n) ∈ X), and s ≤ t stands for s < t ∨ s = t.
Definition 1.3. (L2 -structures). A model for L2 , also called a structure for L2 or an
L2 -structure, is an ordered 7-tuple
M = (NM , SM , +M , ·M , 0M , 1M , <M ) ,
where NM is a set serving as the range of the number variables, SM is a set of subsets of
NM serving as the range of the set variables, +M and ·M are binary operations on NM ,
0M and 1M are distinguished elements of NM , and <M is a binary relation on NM . We
always assume that the sets NM and SM are disjoint and nonempty. Formulas of L2 are
interpreted in M in the obvious way.
“The intended model for L2 is of course the model
(ω, P (ω), +, ·, 0, 1, <)
where ω is the set of natural numbers, P (ω) is the set of all subsets of ω and +, ·, 0, 1, <
are as usual. By an ω-model we mean an L2 -structure of the form
(ω, S, +, ·, 0, 1, <)
where ∅ =
6 S ⊆ P (ω). We sometimes just speak of the ω-model S.” ([1] Section I.2.)
1
1.1
The formal system Z2
Definition 1.4. (second-order arithmetic). The axioms of second-order arithmetic consist
of the universal closure of the following L2 formulas:
1. basic axioms:
• n + 1 6= 0
• m+1=n+1→m=n
• m+0=m
• m + (n + 1) = (m + n) + 1
• m·0=0
• m · (n + 1) = (m · n) + m
• ¬m < 0
• m < n + 1 ↔ (m < n ∨ m = n)
2. induction axiom:
(0 ∈ X ∧ ∀n(n ∈ X → n + 1 ∈ X)) → ∀n(n ∈ X)
3. comprehension scheme:
∃X∀n(n ∈ X ↔ ϕ(n))
where ϕ(n) is any formula of L2 in which X does not occur freely.
In the comprehension scheme, ϕ(n) may contain free variables in addition to n. These
free variables may be referred to as parameters of this instance of the comprehension
scheme.
Remark 1.5. The full second-order induction scheme, i.e., the universal closure of
(ϕ(0) ∧ ∀n(ϕ(n) → ϕ(n + 1))) → ∀nϕ(n) ,
where ϕ(n) is a any formula of L2 , is valid in any model of Definition 1.4.(ii)-(iii).
By second-order arithmetic we mean the formal system in the language L2 consisting
of these axioms, together with all formulas of L2 wich are deducible from those axioms by
means of the usual logical axioms and rules of inference. The formal system of second-order
arithmetic is known as Z2 .
2
2
Subsystems of Z2
By a subsystem of Z2 we mean a formal system in the language L2 each of whose axioms
is a theorem of Z2 . When introducing a new subsystem of Z2 , we shall specify the aixoms
of the system by writing down some formulas of L2 . The axioms are then taken to be the
universal closures of those formulas.
2.1
RCA0
In our designation RCA0 , the acronym RCA stands for recursive comprehension axiom.2
The subscript 0 denotes restricted induction. This means that RCA0 does not include the
full second-order induction scheme as defined in Remark 1.5.3
Let n be a number variable, let t be a numerical term not cointaining n, and let ϕ be
a formula of L2 . We use the following abbreviations:
∀n < t ϕ ≡ ∀n(n < t → ϕ) ,
∃n < t ϕ ≡ ∃n(n < t ∧ ϕ) .
The expressions ∀n < t, ∀n ≤ t, ∃n < t, ∃n ≤ t are called bounded number quantifiers, or
simply bounded quantifiers. A bounded quantifier formula is a formula ϕ such that all of
the quantifiers in ϕ are bounded number quantifiers.
Definition 2.1. (Σ01 and Π01 formulas). An L2 -formula ϕ is said to be Σ01 (respectively
Π01 ) if it is of the form ∃mθ (respectively ∀mθ), where m is a number variable and θ is a
bounded quantifier formula.
Definition 2.2. (∆01 formulas). An L2 formula ϕ is said to be ∆01 if it can be equivalently
written in both classes of Σ01 and Π01 formulas, i.e., we have
ϕ ↔ ψ ↔ η,
where ψ is a Σ01 formula, and η a Π01 formula.
Note that any bounded quantifier formula4 can easily be seen to be ∆01 .
Definition 2.3. (Σ01 and Π01 induction). The Σ01 induction scheme, Σ01 -IND, is the
restriction of the second-order induction scheme to L2 -formulas ϕ(n) which are Σ01 . The
Π01 induction scheme, Π01 -IND, is defined similarly.
It can be shown that in the presence of the basic axioms from Definition 1.4.(i) that
and Π01 -IND are equivalent.
Σ01 -IND
2
Roughly speaking, the set existence axioms of RCA0 are only strong enough to prove the existence of
recursive sets of natural numbers.
3
The same reason applies to all subsystems of Z2 considered in this section.
4
When defining Σ0k and Π0k formulas for general 0 ≤ k ∈ ω, these can be seen as Σ00 or Π00 formulas,
see [1] Definition 1.7.8.
3
Definition 2.4. (∆01 comprehension). The ∆01 comprehension scheme consists of (the
universal closure of) all formulas of the form
∃X∀n(n ∈ X ↔ ϕ(n)) ,
where ϕ is ∆01 , n is any number variable, and X is a set variable which does not occur
freely in ϕ(n).
Definition 2.5. (definition of RCA0 ). RCA0 is the subsystem of Z2 consisting of the
basic axioms 1.4.(i), the Σ01 induction scheme, and the ∆01 comprehension scheme.
An extremely important fact that one can prove within RCA0 , is that each finite set
of natural numbers, i.e., a set of natural numbers for which all its elements are bounded
by some natural number, has a unique code, which is also a natural number. For more
information, see [1] Section II.2.
2.1.1
The number system and primitive recursion
This subsection will be largely based on notes from [1] Section II.3 and II.4, but includes
results that are of much importance for the rest of this thesis.
Our first observation is that within RCA0 , we can define N to be the unique set X
such that ∀n(n ∈ X) by applying ∆01 comprehension to the ∆01 -formula ϕ(n) ≡ 0 = 0,
since for each constant c, the formula c = c is universally valid. It can be shown in RCA0
that the system (N, +, ·, 0, 1, <) is a commutative ordered semiring with cancellation. See
[1] Lemma II.2.1 for a list of some of the basic known arithmetical properties that can be
shown to hold in this system within RCA0 .
Furthermore, we can define the pairing map (i, j) = (i + j)2 + i. It can be proved in
RCA0 that (i, j) = (i0 , j 0 ) ↔ (i = i0 ∧ j = j 0 ). By this, using ∆01 comprehension, we can
prove that for all sets X, Y ⊆ N,5 there exists a set X × Y ⊆ N consisting of all (m, n)
such that m ∈ X and n ∈ Y .
We can introduce total functions from N into N by encoding them as certain sets of
ordered pairs. An important theorem says that the universe of k-ary functions, k ∈ N, is
closed under primitive recursion.
Theorem 2.6. (primitive recursion). The following is provable in RCA0 . Given functions
f : Nk → N and g : Nk+2 → N, there exists a unique h : Nk+1 → N defined by
h(0, n1 , . . . , nk ) = f (n1 , . . . , nk ) ,
h(m + 1, n1 , . . . , nk ) = g(h(m, n1 , . . . , nk ), m, n1 , . . . , nk ) .
Proof. See [1] Theorem II.3.4.
The most important consequence of this theorem is that elementary number theory can
be developed straightforwardly within RCA0 . For instance we can prove the existence of
the exponential function f (m, n) = mn defined by f (m, 0) = 1, f (m, n + 1) = f (m, n) · m.
We can then show that RCA0 proves the validity of some basic ordinary properties
5
The inclusion and equality of set variables can be introduced in the natural way.
4
such as (m1 m2 )n = mn1 mn2 , mn1 +n2 = mn1 mn2 , mn1 n2 = (mn1 )n2 . Also within RCA0 we
can straightforwardly state and prove fundamental results such as unique prime power
factorization. RCA0 may be viewed as a formal version of computable or constructive
mathematics. Moreover, we have the following theorem:
Theorem 2.7. The minimum ω-model of RCA0 is the collection
REC = {X ⊆ ω : X is recursive} .
Proof. See [1] Corollary II.1.8.
The construction of the number system within the context of reverse mathematics is
based on the usual Dedeking/Cauchy construction.
Consider the set N × N of ordered pairs of natural numbers. We define the following
operations and relations on N × N:
• (m, n) +Z (p, q) = (m + p, n + q) ,
• (m, n) −Z (p, q) = (m + q, n + p) ,
• (m, n) ·Z (p, q) = (m · p + n · q, m · q + n · p) ,
• (m, n) <Z (p, q) ↔ m + q < n + p ,
• (m, n) =Z (p, q) ↔ m + q = n + p .
Note that the relations <Z and =Z are expressed by Σ00 formulas. Since the relation =
defines an equivalence relation on the natural numbers, it is easy to see that the relation
=Z defines an equivalence relation on N × N. We define an integer to be an element in
z ∈ N × N of the form
z = min{(m, n) ∈ N × N : (m, n) =Z (a, b)} ,
for some (a, b) ∈ N × N, i.e., z is the N-minimal of its equivalence class. One can prove
within RCA0 that the set of all integers, denoted by Z, exists. If z =N (m, n) ∈ N × N,
then z corresponds to the ordinary integer m − n. It is then possible to accordingly define
the addition +, the subtraction −, the multiplication · and the relation < on Z, as well as
the constants 0 and 1. One can prove within RCA0 some of the basic properties of the
system (Z, +, −, ·, 0, 1, <), such as that it’s an ordered integral domain, Euclidean, and so
on, see [1] Theorem II.4.1.
In a similar manner, we can define within RCA0 the set Q of rational numbers. Let
+
Z be the set of positive integers, i.e.,
Z + := {z ∈ Z : z > 0} .
This set exists within RCA0 since the property z > 0 is expressed by a Σ00 formula. We
define the following operations and relations on the set Z × Z+ :
• (m, n) +Q (p, q) = (m · q + n · p, n · q) ,
5
• (m, n) −Q (p, q) = (m · q − n · p, n · q) ,
• (m, n) ·Q (p, q) = (m · p, n · q) ,
• (m, n) <Q (p, q) ↔ m · q < n · p ,
• (m, n) =Q (p, q) ↔ m · q = n · p .
Again, the relations <Q and =Q are expressed by Σ00 formulas and the relation =Q is an
equivalence relation on Z × Z+ . In an analogous way, we define a rational number to
be the least element of its equivalence class. If q = (m, n), then q corresponds to the
ordinary rational m/n. The set of all rational numbers is denoted by Q and again we may
define +, −, ·, 0, 1 and < on Q accordingly. Within RCA0 , one can show that the system
(Q, +, −, ·, 0, 1, <) is on ordered field, see [1] Theorem II.4.2.
We define an (infinite) sequence of rational numbers to be a function f : N → Q and
denote such a sequence by hqn : n ∈ Ni or simply by hqn i, where qn = f (n). A real number
is defined to be a sequence han i of rational numbers such that
∀n∀i(|an − an+i | ≤ 2−n ) ,
where | · | denotes the absolute value
| · | : Q → Q+ : q 7→

q
if q ≥ 0
−q
if q < 0
.
We use R informally6 to denote the set of all real numbers, i.e., ‘x ∈ R’ is just a notation
for ‘x is a real number’. The set Q can be embedded into R by identifying q ∈ Q with
the sequence hqn : n ∈ Ni for which qn = q for all n ∈ N. Let x = hpn : n ∈ Ni and
y = hqn : n ∈ Ni be two real numbers. Their sum is defined as
x + y = hpn+1 + qn+1 : n ∈ Ni ,
and their product as
x · y = hpk+n + qk+n : n ∈ Ni ,
where k is defined as
k := min{m ∈ N : 2m ≥ |p0 | + |q0 | + 2} .
It is easy to check that x + y and x · y indeed again define real numbers. Furthermore, we
define the following relations on R:
• x = y ↔ ∀n(|pn − qn | ≤ 2−n+1 ) ,
• x ≤ y ↔ ∀n(pn ≤ qn + 2−n+1 ) ,
• x < y ↔ y x ↔ ∃n(qn > pn + 2−n+1 ) .
6
The set R does not exist within RCA0 , since RCA0 is limited to the language L2 of second-order
arithmetic.
6
Note that the relations = and ≤ on R are expressed by Π01 formulas, whereas the relations
< and 6= are expressed by Σ01 formulas. It is easy to check within RCA0 that the system
(R, +, −, 0, 1, <, =) obeys all the axioms for an ordered Abelian group. Furthermore, one
can show that the real number system (R, +, −, ·, 0, 1, <, =) obeys all the axioms of an
Archimedean ordered field, see [1] Theorem II.4.5.
An infinite sequence of real numbers is defined to be a doubly indexed sequence of
rational numbers hqmn : m, n ∈ Ni such that for each m, hqmn : n ∈ Ni is a real number.
Such a sequence is also denoted by hxm : m ∈ Ni, where xm = hamn : n ∈ Ni. We write
RN for the informal set of infinite sequences of real numbers. We say that the sequence
hxm : m ∈ Ni converges to x, where x ∈ R, if
∀ ∈ R+ ∃m ∀i(|x − xm+i | < ) ,
and we denote limm xm = x.
Remark 2.8. Our construction of the number system is based on notes from [1] Section
II.4. The main reason why we included this construction in this thesis, is to make the
reader aware why relations such as x < y and x 6= y, with x, y ∈ R, are expressed by
Σ01 formulas. This fact is of much importance for the rest of the thesis and will be used
quite a lot to be able to construct specific open sets and their complements in complete
separable metric spaces, see also Theorem 2.14.
2.1.2
Concepts of analysis and topology
In this subsection, we will focus on some analytical and topological concepts that can be
developed within RCA0 and will play important roles in this thesis. Nevertheless, many
basic concepts concerning mathematical logic and countable algebra can also be adequately
defined within RCA0 .
Two important definitions made in RCA0 are those of a complete separable metric
space and of a compact metric space:
Definition 2.9. (complete separable metric spaces, [1] Definition II.5.1). A (code for a)
b is defined in RCA0 to be a nonempty set A ⊆ N together
complete separable metric space A
with a sequence of real numbers d : A × A → R such that d(a, a) = 0, d(a, b) = d(b, a) ≥ 0,
b is a sequence x = hak : k ∈ Ni
and d(a, b) + d(b, c) ≥ d(a, c) for all a, b, c ∈ A. A point of A
b to mean that
of elements of A, such that ∀i ∀j(i < j → d(ai , aj ) ≤ 2−i ). We write x ∈ A
b
x is a point of A.
b we can define
If x = hak : k ∈ Ni and y = hbk : k ∈ Ni are two points of A,
d(x, y) = limk d(ak , bk ). We define x = y to mean that d(x, y) = 0. Each a ∈ A is of
b By definition, it now easily follows
course identified with the point xa = ha : k ∈ Ni ∈ A.
b Indeed, for x = hak : k ∈ Ni, it holds that
that the countable set A is dense in A.
b as separable. To justify
d(x, an ) = limk d(ak , an ) ≤ 2−n . This justifies our designation of A
b is complete in the following sense:
our designation of of “complete”, we note that A
Theorem 2.10. The following is provable in RCA0 . If hxn : n ∈ Ni is a sequence of points
b and there exists a sequence of real numbers hrn : n ∈ Ni with limn rn = 0 and such
in A,
7
that ∀m ∀n(m < n → d(xn , xm ) ≤ rm ), then hxn : n ∈ Ni is convergent, i.e., there exists a
b such that x = limn xn .
point x ∈ A
Proof. See [1] Exercise II.5.2.
Note that this theorem also justifies why we are able to define d(x, y) = limk d(ak , bk ),
where x = hak : k ∈ Ni and y = hbk : k ∈ Ni are points in a complete separable metric
b i.e., we can see R as the completion of the rationals
b Moreover, we can let R = Q,
space A.
under the metric dQ : Q × Q → R with dQ (p, q) = |p − q| for p, q ∈ Q. Since for every
b 7 hd(ak , bk ) : k ∈ Ni defines a sequence of points in Q
b for
k ∈ N we have d(ak , bk ) ∈ Q,
which for all m, n ∈ N, m < n
|d(an , bn ) − d(am , bm )| ≤ |d(an , bn ) − d(an , bn−1 )| + . . . + |d(an , bm+1 ) − d(an , bm )|
+ |d(an , bm ) − d(an−1 , bm )| + . . . + |d(am+1 , bm ) − d(am , bm )|
≤ d(bn , bn−1 ) + . . . + d(bm+1 , bm ) + d(an , an−1 ) + . . . d(am+1 , am )
≤ 2 · (2−(n−1) + . . . + 2−m ) = 2−(n−2) + . . . + 2−(m−1) ≤ 2−(m−2) .
so that limk d(ak , bk ) indeed exists within RCA0 . Hereby, we note that for a real number
r = hqn : n ∈ Ni, one can explicitly define |r| = h|qn | : n ∈ Ni. The fact that this is indeed
again a real number, follows easily from the triangle inequality.
Definition 2.11. (compactness, [1] Definition III.2.3). The following definition is made
b such that there
in RCA0 . A compact metric space is a complete separable metric space A
exists an infinite sequence of finite sequences
hhxij : i ≤ nj i : j ∈ Ni,
b,
xij ∈ A
b and j ∈ N there exists i ≤ nj such that d(xij , z) < 2−j .
such that for all z ∈ A
We can define the concepts of open and closed sets in and continuous functions between
complete separable metric spaces:
Definition 2.12. (open and closed sets, [1] Definitions II.5.6 & II.5.12). Within RCA0 ,
b be a complete separable metric space. A (code for an) open set U in A
b is a set
let A
+
b is said to belong to U , denoted by x ∈ U , if
U ⊆ N × A × Q . A point x ∈ A
∃n ∃a ∃r (d(x, a) < r ∧ (n, a, r) ∈ U ) .
b is the complement of an open set in A.
b I.e., a code for a closed set C is
A closed set in A
the the same thing as a code for an open set U , but we define x ∈ C iff x ∈
/ U.
We regard (a, r) ∈ A × Q+ as a code for the basic open ball B(a, r) consisting of all
b such that d(x, a) < r. The idea of the preceding definition is that U encodes
points x ∈ A
the open set which is the union of the open balls B(a, r) such that ∃n((n, a, r) ∈ U ). By
these definitions, one can prove some of the basic properties of open and closed sets, such
as:
7
b coincide
The definitions of elements of R and elements of Q
8
Theorem 2.13. The following is provable in RCA0 . Suppose C is a closed set in a
b and that hxn : n ∈ Ni is a sequence of points in C. If
complete separable metric space A
limn→∞ xn = x, then x ∈ C.
Proof. Let U denote the code for the open complement of C, such that y ∈ C iff y ∈
/ U,
i.e., iff
∀m ∀a ∀r(d(y, a) ≥ r ∨ (m, a, r) ∈
/ U) .
Suppose x ∈
/ C. By definition this is the case iff x ∈ U , i.e., iff
∃m ∃a ∃r (d(x, a) < r ∧ (m, a, r) ∈ U ) .
Now take such (m, a, r) ∈ U with d(x, a) < r. Since xn ∈ C, we must have d(xn , a) ≥ r
for all n ∈ N. Now let N ∈ N be such that n > N implies d(xn , x) < (r − d(x, a))/2 and
note that d(x, a) < (d(x, a) + r)/2. Hence, we find for n > N
d(xn , a) ≤ d(xn , x) + d(x, a) <
r − d(x, a) r + d(x, a)
+
= r,
2
2
a contradiction. Hence, we must have x ∈ C. See also Figure 1.
Figure 1: If x ∈ B(a, r) and xn → x, then for n large enough, we must always have
xn ∈ B(a, r).
Note that many examples of open sets, and hence, their closed complements, can be
shown to exist within RCA0 . This follows from the following theorem.
Theorem 2.14. The following is provable in RCA0 . Let for every n ∈ N, ϕ(x, n) be a Σ01
b be a complete separable metric space. Assume that for all n ∈ N and
formula, and let A
b (x = y) ∧ ϕ(x, n) implies ϕ(y, n). Then there exists a sequence of (codes for)
x, y ∈ A,
b for every n ∈ N, such that for all x ∈ A,
b x ∈ Un if
open sets hUn : n ∈ Ni with Un ⊆ A
and only if ϕ(x, n). A similar statement holds for a Σ01 formula ϕ(x) and the existence of
b such that x ∈ U ↔ ϕ(x).
the open set U ⊆ A
9
Proof. See [1] Lemma II.5.7, where a proof for the second statement is given.
b and E
b
Definition 2.15. (continuous functions, [1] Definition II.6.1). Within RCA0 , let A
b
be complete separable metric spaces. A (code for a) continuous partial function Φ from A
+
+
b
to E is a set of quintuples Φ ⊆ N × A × Q × E × Q with certain properties. We write
(a, r)Φ(b, s) as an abbreviation for ∃n((n, a, r, b, s) ∈ Φ). The required properties are
1. if (a, r)Φ(b, s) and (a, r)Φ(b0 , s0 ), then d(b, b0 ) ≤ s + s0 ;
2. if (a, r)Φ(b, s) and d(a, a0 ) + r0 < r, then (a0 , r0 )Φ(b, s) ;
3. if (a, r)Φ(b, s) and d(b, b0 ) + s < s0 , then (a, r)Φ(b0 , s0 ) .
Figure 2: Intuitively, the first property of
a continuous functions states that when
B(a, r) gets mapped to both B(b, s) and
B(b0 , s0 ), then these balls ‘should’ overlap.
One does need to take caution with this
interpration, since in general complete separable metric spaces, d(b, b0 ) ≤ s + s0 and
B(b, s) ∩ B(b0 , s0 ) 6= ∅ are’nt always equivalent. See also Remark 4.16.
Figure 3: The second property of a continuous functions states that when B(a, r)
gets mapped to B(b, s), and we have
B(a0 , r0 ) ⊆ B(a, r), then also B(a0 , r0 ) gets
mapped to B(b, s).
Figure 4: The third property of a continuous functions states that when B(a, r) gets
mapped to B(b, s) and B(b, s) ⊆ B(b0 , s0 ), then B(a, r) gets also mapped to B(b0 , s0 ).
10
“The idea of the definition is that Φ encodes a partially defined, continuous function
b to E.
b Intuitively, (a, r)Φ(b, s) is a piece of information to the effect that φ(x)
φ from A
is contained within the closure of B(b, s) whenever x ∈ B(a, r), provided φ(x) is defined.
b is said to belong to the domain of φ, abbreviated x ∈ dom(φ), provided
A point x ∈ A
the code Φ of φ contains sufficient information to evaluate φ at x. This means that for
all > 0 there exists (a, r)Φ(b, s) such that d(x, a) < r and s < . If x ∈ dom(φ), we
b such that d(y, b) ≤ s for all (a, r)Φ(b, s) with
define the value φ(x) to be the point y ∈ E
d(x, a) < r. If x ∈ dom(φ), then within RCA0 , it is possible to show that φ(x) exists and
b We write φ(x) is defined to mean that x ∈ dom(φ).
is unique up to equality of points in E.
b
b We write φ : A
b→E
b
We say that φ is totally defined on A if φ(x) is defined for all x ∈ A.
b to E.”
b ([1] Section II.6.)
to mean that φ is a continuous, totally defined function from A
Note that this definition implies some of the usual properties of continuous functions,
such as:
b→E
b is a continuous function between two complete separable
Theorem 2.16. If f : A
b
metric spaces and limn→∞ xn = x in dom(f ), then limn→∞ f (xn ) = f (x) in E.
Proof. Take > 0. We need to show that there exists an N ∈ N such that n > N implies
dE (f (xn ), f (x)) < . Since x ∈ dom(f ), there exists (a, r)Φ(b, s) such that dA (x, a) < r and
s < /2. By definition, we then have dE (f (x), b) ≤ s < /2. Take N ∈ N such that n > N
implies dA (xn , x) < (r − dA (x, a))/3. Now let q ∈ Q+ be such that q < (r − dA (x, a))/3.
Then for each n > N , there exists an an ∈ A such that dA (an , xn ) < q, and we find
dA (a, an ) + q ≤ dA (x, a) + dA (x, xn ) + dA (xn , an ) + q < dA (x, a) + 3 ·
r − dA (x, a)
= r.
3
so that Property 2. of Definition 2.15 implies (an , q)Φ(b, s). Since dA (xn , an ) < q and
xn ∈ dom(f ), we again have by definition dE (f (xn ), b) ≤ s < /2, hence
dE (f (x), f (xn )) ≤ dE (f (x), b) + dE (f (xn ), b) < ,
which ends our proof. See also Figure 5 for an illustration.
Figure 5: If xn → x ∈ B(a, r) 7→f B(b, s), then for n large enough, xn is always contained
in a ball B(an , q) ⊆ B(a, r) that also gets mapped onto the ball B(b, s) by f .
11
These concepts will be used when performing the reverse mathematics of our version of
Brouwer’s fixed point theorem in Chapter 3. When formulating the Browder-Göhde-Kirk
fixed point theorem in Chapter 4, we will make use of the concept of a separable Banach
space:
Definition 2.17. (fields, [1] Definition II.9.1). A countable field K consists of a set |K| ⊆ N
together with binary operations +K , ·K , a unary operation −K (symbolizing the additive
inverse) and distinguished elements 0K , 1K , such that the system (|K|, +K , −K , ·K , 0K , 1K )
obeys the usual field axioms.
Definition 2.18. (vector spaces, [1] Section II.10). Let K be a countable field. Within
RCA0 , a countable vector space A over K consists of a set |A| ⊆ N together with operations
+ : |A| × |A| → |A| and · : |K| × |A| → |A| and a distinguished element 0 ∈ |A|, such that
the system (|A|, +, ·, 0) satisfies the usual axioms for a vector space over K.
Definition 2.19. (separable Banach spaces, [1] Definition II.10.1). Within RCA0 , we
b to consist of a countable vector space A
define a (code for a) separable Banach space A
over the rational field Q together with a sequence of real numbers || · || : A → R satisfying
1. ||q · a|| = |q| · ||a|| for all q ∈ Q and a ∈ A ;
2. ||a + b|| ≤ ||a|| + ||b|| for all a, b ∈ A.
b is defined to be a sequence hak : k ∈ Ni of elements of A such that
A point of A
||ak − ak+1 || ≤ 2−k−1 for all k ∈ N.
Note that for any a ∈ A
0 = ||0|| = ||a − a|| ≤ ||a|| + || − a|| = 2||a|| ↔ 0 ≤ ||a|| ,
so that ∀a ∈ A, ||a|| ≥ 0 It is possible to show within RCA0 that one can extend the
b → R, + : A
b×A
b→ A
b and
operators || · ||, + and · to continuous operators || · || : A
b→A
b by taking the usual limits. We define x = y to mean that ||x − y|| = 0. One
· : R×A
b enjoys the usual properties of a normed vector space over R. As usual,
can show that A
we define a pseudometric on A by d(a, b) = ||a − b||, for all a, b ∈ A. By this, it is easily
b is the complete separable metric space which is the completion of A under d.
seen that A
2.2
WKL0
We note that concepts like the set of all (codes for) finite sequences Seq or N<N , the length
lh of a sequence, the concatenation _ of two sequences, being an initial segment of another
sequence (denoted by ⊆) and the set 2<N of all finite sequences of 0’s and 1’s can be
perfectly defined within RCA0 .
Definition 2.20. (König’s lemma, [1] Definition 2.18). The following definitions are made
in RCA0 . A tree is a set T ⊆ N<N which is closed under initial segment, i.e.,
∀σ∀τ ((σ ∈ N<N ∧ σ ⊆ τ ∧ τ ∈ T ) → σ ∈ T ) .
12
We say that T is finitely branching if each element of T has only finitely many immediate
successors, i.e.
∀σ(σ ∈ T → ∃n∀m(σ _hmi ∈ T → m < n)) .
A path through T is a function g : N → N such that g[n] := hg(0), g(1), . . . , g(n − 1)i ∈ T
for all n ∈ N. König’s lemma is the assertion that every infinite, finitely branching tree T
has at least one path.
Definition 2.21. (weak König’s lemma, [1] Definition I.10.1). The following defintions
are made within RCA0 . We use {0, 1}<N or 2<N to denote the full binary tree, i.e., the
set of (codes for) finite sequences of 0’s and 1’s. Weak König’s lemma is the following
statement:
Every infinite subtree of 2<N has an infinite path.
WKL0 is defined to be the subsystem of Z2 consisting of RCA0 plus weak König’s lemma.
Hence, the subsystem RCA0 of Z2 is included in WKL0 .
Remark 2.22. It is clear that an infinite subtree of the full binary tree is an infinite,
finitely branching tree. Hence, as one could already suspect, König’s lemma implies weak
König’s lemma.
2.3
ACA0
The acronym ACA stands for arithmetical comprehension axiom. This is because ACA0
contains axioms asserting the existence of any set which is arithmetically definable from
given sets.
Definition 2.23. (arithmetical formulas). A formula of L2 is said to be arithmetical if it
contains no set quantifiers, i.e., all of the quantifiers appearing in the formula are number
quantifiers.
Definition 2.24. The arithmetical comprehension scheme is the restriction of the comprehension scheme 1.4.(iii) to arithmetical formulas ϕ(n).
ACA0 is the subsystem of Z2 whose axioms are the arithmetical comprehension scheme,
the induction axiom 1.4.(ii), and the basic axioms 1.4.(i).
An easy consequence of the arithmetical comprehension scheme and the induction
axiom is the arithmetical induction scheme:
(ϕ(0) ∧ ∀n(ϕ(n) → ϕ(n + 1))) → ∀nϕ(n) ,
for all L2 -formulas ϕ(n) which are arithmetical.
Remark 2.25. Since all Σ01 and Π01 formulas are of course arithmetical, Definition 2.5
implies that the subsystem ACA0 of Z2 includes RCA0 .
Actually, also the subsystem WKL0 is included in ACA0 . In order to show this, we
would need to prove that ACA0 implies weak König’s lemma over RCA0 . Considering
Remark 2.22, there even holds a stronger result:
Theorem 2.26. ACA0 is equivalent to König’s lemma over RCA0 .
Proof. See [1] Theorem III.7.2.
13
2.4
ATR0
The acronym ATR stands for arithmetical transfinite recursion. We start with the
definition of a countable well-ordering.
Definition 2.27. (countable well-ordering [1] Definition I.6.1). Within RCA0 , we define
a countable linear ordering to be a structure (A, <A ) , where A ⊆ N and <A ⊆ A × A is a
strict linear ordering of A, i.e., <A is transitive and, for all a, b ∈ A, exactly one of the
relations a = b or a <A b or b <A a holds. The countable linear ordering A, <A is called
a countable well-ordering if there is no sequence han : n ∈ Ni of elements in A such that
an+1 <A an for all n ∈ N.
Definition 2.28. (arithmetical transfinite recursion, [1] Definition I.11.1). Consider an
arithmetical formula θ(n, X) with a free number variable n and a free set variable X. Note
that θ(n, X) may also contain parameters, i.e., additional free number and set variables.
Fixing these parameters, we may view θ as an “arithmetical operator” Θ : P(N) → P(N),
defined by
Θ(X) = {n ∈ N : θ(n, X)} .
Now let (A, <A ) be any countable well-ordering, and consider the set Y ⊆ N obtained by
transfinitely iterating the operator Θ along (A, <A ). This set Y is defined by the following
conditions: Y ⊆ N × A and, for each a ∈ A, Ya = Θ(Y a ), where Ya = {m : (m, a) ∈ Y }
and Y a = {(n, b) : n ∈ Yb ∧ b <A a}. Thus, for each a ∈ A, Y a is the result of iterating
Θ along the initial segment of (A, <A ) up to but not including a, and Ya is the result of
applying Θ one more time. Finally, arithmetical transfinite recursion is the axiom scheme
asserting that such a set Y exists, for every arithmetical operator Θ and every countable
well-ordering (A, <A ).
We define ATR0 to consist of ACA0 plus the scheme of arithmetical transfinite recursion.
Hence, ACA0 is trivially included in the subsystem ATR0 of Z2 .
2.5
Π11 -CA0
Definition 2.29. (Π11 formulas). A formula ϕ is said to be Π11 if it is of the form ∀X θ ,
where X is a set variable and θ is an arithmetical formula. A formula ϕ is said to be Σ11 if
it is of the form ∃X θ, where X is a set variable and θ is an arithmetical formula.
Definition 2.30. (Π11 comprehension). Π11 -CA0 is the subsystem of Z2 whose axioms
are the basic axioms, the induction axiom, and the comprehension scheme restricted to
L2 -formulas ϕ(n) which are Π11 . Thus we have the universal closure of
∃X ∀n(n ∈ X ↔ ϕ(n))
for all Π11 formulas ϕ(n) in which X does not occur freely.
It would be possible to introduce the system Σ11 -CA0 , but this would be superfluous,
because one can show that Σ11 -CA0 and Π11 -CA0 are equivalent, i.e., they have the same
theorems.
14
It turns out that ATR0 is equivalent to several theorems of ordinary mathematics
which are provable in Π11 -CA0 but not in ACA0 , which implies that ATR0 is intermediate
between ACA0 and Π11 -CA0 .
We mentioned the subsystems ATR0 and Π11 -CA0 just for completeness. They will not
play any important roles in the further development of this thesis.
15
3
Reverse mathematics
In this chapter we shall discuss the subject of reverse mathematics. We shall formulate
the Main Question, and, as an example, we shall include Simpson’s analysis of the reverse
mathematics of Brouwer’s fixed point theorem given in [1] Chapter IV.7. However, we
shall adjust some of the theorems and proofs and stick to the case where the domain is
the unit square [0, 1]2 .
3.1
The Main Question
In the previous chapter we have discussed five specific natural subsystems of Z2 , where
each system is included in the next one. In general, for a given theorem, one can wonder
which is the weakest natural subsystem of Z2 in which we can prove this theorem. This is
the Main Question of reverse mathematics:
Given a theorem τ of ordinary mathematics, what is the weakest natural subsystem
S(τ ) of Z2 in which τ is provable?
Suprisingly, S(τ ) often turns out to be one of the five specific subsystems of Z2 which
we shall now list as
• S1 =RCA0 ,
• S2 =WKL0 ,
• S3 =ACA0 ,
• S4 =ATR0 ,
• S5 = Π11 -CA0 ,
in order of increasing ability to accommodate ordinary mathematical practice.
Our method for establishing results of the form S(τ ) = Sj , 2 ≤ j ≤ 5 is based on the
empirical phenomenon that when the theorem is proved from the right axioms, the axioms
can be proved from the theorem. More specifically, let τ be an ordinary mathematical
theorem which is not provable in the weak base theory S1 = RCA0 . Then very often, τ
turns out to be equivalent to Sj for some j = 2, 3, 4 or 5. This equivalence is provable in
Si for some i < j, usually i = 1.
3.2
3.2.1
Brouwer’s fixed point theorem for the unit square
The unit square in R2 .
b where d : Q×Q → R is given by d(q, q 0 ) = |q−q 0 |.
R is the complete separable metric space Q
In general, given n complete separable metric spaces A1 , . . . , An , one defines the n-fold
cartesian product
A = A1 × . . . × An = {ha1 , . . . , an i : ai ∈ Ai }
16
and d : A × A → R by
d(ha1 , . . . , an i, hb1 , . . . , bn i) =
p
d1 (a1 , b1 )2 + . . . + dn (an , bn )2
b is a complete separable space, that the points of
and one can prove within RCA0 that A
bi , ∀i ≤ n, and that
b can be identified with the finite sequences hx1 , . . . xn i, with xi ∈ A
A
b is also given by
under this identification, the metric on A
p
d(hx1 , . . . , xn i, hy1 , . . . , yn i) = d1 (x1 , y1 )2 + . . . dn (xn , yn )2 ,
see also [1] Example II.5.4.
b
Hence, if the set E is given by the cartesian product Q × Q, then
pE forms the separable
2
Banach space R under the identification ||(a1 , b1 ) − (a2 , b2 )|| = |a1 − a2 |2 + |b1 − b2 |2 ,
and all relevant operations can within RCA0 be extended to R2 . This means that the
b is given by the standard Euclidean norm.
norm in R2 = E
Given hx, yi ∈ R2 , |x| and |y| represent real numbers, and we define the Σ01 formulas
ϕ1 (hx, yi) ≡ |x| > 1 ;
ϕ2 (hx, yi) ≡ |x| < 0 ;
ϕ3 (hx, yi) ≡ |y| > 1 ;
ϕ4 (hx, yi) ≡ |y| < 0 .
Note that these formulas are indeed Σ01 , see Remark 2.8. Now suppose hx, yi = hu, vi as
points in R2 . Under the given identification, this means we have x = u and y = v as points
in R. Hence we have ϕi (hx, yi) → ϕi (hu, vi) for i = 1, 2, 3, 4. Theorem 2.14 implies the
existence of (codes for) the open sets
U1 = {(hx, yi) ∈ R2 : |x| > 1} ;
U2 = {(hx, yi) ∈ R2 : |x| < 0} ;
U3 = {(hx, yi) ∈ R2 : |y| > 1} ;
U4 = {(hx, yi) ∈ R2 : |y| < 0} .
Using these codes, we can define (a code for) the open set
U = U1 ∪ U2 ∪ U3 ∪ U4 .
Now we also have (a code for) the closed complement C = [0, 1]2 of U . Hence, the unit
square can be defined as a closed set within RCA0 .
3.2.2
Proving that Brouwer’s fixed point theorem for the unit square is equivalent to WKL0 over RCA0 .
As an example, we shall hereby include Simpson’s analysis of the reverse mathematics
of Brouwer’s fixed point theorem, given in [1] Section IV.7. The ordinary mathematical
statement is as follows:
17
Theorem 3.1. (Brouwer’s fixed point theorem). Let C be the convex hull of a nonempty
finite set of points in Rn , n ∈ N. Then every continuous function f : C → C has a fixed
point.
However, we shall restrict the analysis to the case where C is the unit square in R2 ,
i.e., we shall perform the reverse mathematics of the following theorem:
Theorem 3.2. (Brouwer’s fixed point theorem for the unit square). Let C be the
unit square in the separable Banach space R2 . Then for any continuous function
f : C → C, there is a point x ∈ [0, 1]2 such that f (x) = x.
We shall prove that Theorem 3.19 is equivalent to WKL0 over RCA0 . Our methods
will be comepletely based on [1] Section IV.7, but simplified to the two-dimensional case
and with added details and illustrations.
Definition 3.3. (affinely independent points). The following definition is made in RCA0 .
Let n ∈ N, n ≥ 1. Suppose for k ∈ N, s0 , . . . , sk−1 and sk are points in the separable
Banach space Rn . Then these points are called affinely independent, if for all real numbers
α0 , . . . , αk , we have that
k
k
X
X
αi si = 0 ∧
αi = 0
i=0
i=0
implies αi = 0 for all i = 0, . . . , k.
Theorem 3.4. The following is provable in RCA0 . Suppose s0 , . . . , sk are affinely independent points in the separable Banach space Rn , n ≥ 1. Then k ≤ n.
Proof. W.l.o.g. we may assume k 6= 0. Then for i = 1, . . . , k, we can define
vi := si − s0 .
It is easy to see that si 6= sj for all 0 ≤ i < j ≤ k, so that the vi ’s are all different nonzero
vectors in Rn . Suppose there exist real numbers α1 , . . . , αk , such that
α1 v1 + . . . + αk vk = 0 ∈ Rn .
Defining
α0 = −
k
X
αi ,
i=1
this means that
k
X
αi si = 0 ∧
i=0
k
X
αi = 0 ,
i=0
so that αi = 0 for all i = 0, . . . k. Hence, v1 , . . . , vk are linearly independent vectors in
Rn . Since RCA0 is strong enough to develop the basics of real linear algebra, including
Gaussian elimination ([1] Exercise II.4.11), we may conclude that k ≤ n.
18
Definition 3.5. (convex hull). The following definition is made in RCA0 . Suppose that
n ∈ N, n ≥ 1, and s0 , . . . , sk ∈ Rn . We define
)
(
k
X
L = hq0 , . . . , qk i ∈ Qk+1 : q0 , . . . , qk ∈ [0, 1] ∧
qi = 1 ,
i=0
and the sequence of reals
k
k
X
X
d : L × L → R : (hq0 , . . . , qk i, hp0 , . . . , pk i) 7→ q i si −
pi si .
i=0
i=0
b called the
Then L together with d forms a (code for a) complete separable metric space L,
convex hull of the points s0 , . . . , sn .
Theorem 3.6. The following is provable in RCA0 . Suppose that n ∈ N, n ≥ 1, and
b of
s0 , . . . , sn are affinely independent points in Rn . Then the points in the convex hull L
s0 , . . . , sn are in a one-to-one correspondence with the points in the informal set
!)
(
n
n
X
X
αi si ∧
αi = 1
,
S = x ∈ Rn : ∃α0 , . . . , αn ∈ [0, 1]R x =
i=0
i=0
given by
k→∞
hρk = hρk0 , . . . , ρkn i : k ∈ N, ρk ∈ Li −−−→ ρ
←→
b
L
n
X
i=0
lim ρki si ∈ S .
k
Moreover, this correspondence preserves distances.
Proof. It is easy to see that this correspondence will preserve distances, since for sequences
hρk = hρk0 , . . . , ρkn i : k ∈ N, ρk ∈ Li, hµk = hµk0 , . . . , µkn i : k ∈ N, µk ∈ Li, we have
(considering all terms exist),
d lim ρk , lim µk = lim d(ρk , µk )
k
k
k
n
n
X
X
= lim ρki si −
µki si k i=0
i=0
n
n
X
X
= lim ρki si −
lim µki si .
k
k
i=0
i=0
Now suppose
k→∞
b.
hρk = hρk0 , . . . , ρkn i : k ∈ N, ρk ∈ Li −−−→ ρ ∈ L
b
L
Then ρ is a sequence hqk = hqk0 ,P
. . . , qkn i : k ∈ N, qk ∈ Li such that d(qk , ql ) ≤ 2−l for
all k, l ∈ N, with k ≥ l. Hence, h ni=0 qki si : k ∈ Ni converges in Rn to some point x by
Theorem 2.10. As in the proof of Theorem 3.4, we know that the points vi := si − s0 , with
19
1 < i ≤ n, are linearly independent in Rn , so that by Gaussian elimination there exists a
unique n-tuple hα1 , . . . , αn i ∈ Rn , such that
x − s0 =
n
X
αi vi .
i=1
for k ∈ N, we define
xk :=
n
X
qki si .
i=0
Since
Pn
i=0 qki
= 1 for all k ∈ N, we have
xk − s 0 =
n
X
qki (si − s0 ) =
i=1
n
X
qki vi ,
i=1
so that
0 = lim(x − xk )
k
= lim ((x − s0 ) − (xk − s0 ))
k
= lim
k
n
X
n X
(αi − qki )vi =
i=1
αi − lim qki vi ,
k
i=0
which implies that limk qki = αi for all i = 1, . . . , n. Since qki ∈ [0, 1] for all k ∈ N, i ≤ n,
clearly αi ∈ [0, 1]R for i = 1, . . . , n. Moreover, defining
α0 := 1 −
n
X
αi ,
i=1
we have
α0 = 1 −
n
X
i=1
lim qki = 1 − lim
k
k
n
X
qki = lim 1 −
k
i=1
n
X
i=1
!
qki
= lim q0k .
k
so that we also have α0 ∈ [0, 1]R , and we find
x=
n
X
αi si ∧
n
X
i=0
αi = 1 .
i=0
Defining
x0k
:=
n
X
ρki si ,
i=0
x0k ||
then since limk ||xk −
= limk d(qk , ρk ) ≤ limk d(qk , ρ) + limk d(ρ, ρk ) = 0, we find in an
analogous way that limk qki = limk ρki for all i ≤ n.
20
P
Conversely, let α0 , . . . , αn ∈ [0, 1]R be such that ni=0 αi = 1. Suppose first 0 6= αi 6= 1
for all i ≤ n. If αi = hqin : n ∈ N, qin ∈ Qi, then there exists N ∈ P
N, such that whenever
k ∈ N, k ≥ N , we have qik , ∈]0, 1[Q , for all i = 1, . . . , n, and 1 − ni=1 qik ∈]0, 1[Q . Now
for every k ∈ N, we define
*
+
n
X
ρk := 1 −
qi,N +k , q1,N +k , . . . , qn,N +k ∈ L
i=1
xk :=
1−
n
X
!
qi,N +k
s0 +
i=1
n
X
qi,N +k si ,
i=1
then for all k, l ∈ N, k ≥ l,
d(ρk , ρl ) = ||xk − xl ||
n
n
X
X
= (qi,N +k − qi,N +l )si +
(qi,N +l − qi,N +k )s0 i=1
≤2
n
X
i=1
|qi,N +k − qi,N +l | max{||s0 ||, . . . , ||sk ||}
i=1
l→∞
≤ n · 2−(N +l)+1 max{||s0 ||, . . . , ||sn ||} −−−→ 0 ,
b by Theorem 2.10. Now if αl = 1
so that the sequence hρk : k ∈ Ni converges to some ρ ∈ L
for some l ∈ {0, . . . , n}, we clearly have
h0, . . . , 0,
b.
1
, 0, . . . , 0i ∈ L
|{z}
l-th place
Finally, suppose αl = 0 for some l ∈ {0, . . . , k}. W.l.o.g., we may assume α0 = . . . = αj = 0
for j < k, and αj+1 , . . . , αk ∈]0, 1[R . Defining
+
*
n
X
ρk := 0, . . . , 0, 1 −
qi,N +k , qj+1,N +k , . . . , qn,N +k ∈ L ,
i=j+1
for N large enough, in an analogous way as in the first case, one concludes that there
b such that ρk →k ρ.
exists ρ ∈ L
Definition 3.7. (triangle, [1] Definition IV.7.1). The following definition is made in RCA0 .
A triangle S in R2 is the convex hull of three affinely independent points s0 , s1 , s2 ∈ R2 ,
called the vertices of S.
Definition 3.8. (diameter) The following definition is made in RCA0 . Let S be a triangle
in R2 , determined by the vertices s0 , s1 , s2 ∈ R2 . The diameter of S is then defined as
diam(S) := max{||s0 − s1 ||, ||s0 − s2 ||, ||s1 − s2 ||} .
Note that diam(S) > 0.
21
Theorem 3.9. The following is provable in RCA0 . Let S be a triangle in R2 . Then S is
a compact metric space.
Proof. Suppose S is determined by the vertices s0 , s1 , s2 ∈ R2 . Let N ∈ N be any natural
number such that 2N > max{||s0 ||, ||s1 ||, ||s2 ||}, and take j ∈ N. By Σ00 comprehension,
we may define the set
Aj := hq0 , q1 , q2 i ∈ Q3 : ∃i ≤ 2j+N +3 , ∃k ≤ 2j+N +3 − i
q0 = i · 2−j−N −3 ∧ q1 = k · 2−j−N −3 ∧ q2 = 1 − (i + k)2−j−N −3 .
P j+N +3
For all n ∈ N, n ≤ 2m=0 (2j+N +3 − m) = |Aj |, we recursively define

minN Aj
if n = 0 ,
xnj =
min {x ∈ A : ∀k < n(x 6= x )} if 0 < n ≤ P2j+N +3 (2j+N +3 − m) ,
N
j
nj
m=0
D
E
P2j+N +3 j+N +3
i.e., xnj : n ≤ m=0
(2
− m) is a finite enumeration of all elements in Aj , j ∈ N.
We claim that the infinite sequence of finite sequences
**
+
+
+3
2j+N
X
xnj : n ≤
2j+N +3 − m
:j∈N ,
m=0
attests to the definition of compactness. Hence, suppose x ∈ S and j ∈ N. By Theorem
3.6, we may represent x as
x ∼ α0 s0 + α1 s1 + α2 s2 ∈ R2 ,
with α0 , α1 , α2 ∈ [0, 1]R and α0 + α1 + α2 = 1, and hq0 , q1 , q2 i ∈ Aj as
hq0 , q1 , q2 i ∼ i · 2−j−N −3 s0 + k · 2−j−N −3 s1 + 1 − (i + k)2−j−N −3 s2 ∈ R2 ,
for some i ≤ 2j+N +3 , k ≤ 2j+N +3 − i, and we have
d(x, hq0 , q1 , q2 i) = α0 − i · 2−j−N −3 s0 + α1 − k · 2−j−N −3 s1
+ i · 2−j−N −3 − α1 + k · 2−j−N −3 − α2 s2 ≤ 2 α0 − i · 2−j−N −3 + α1 − k · 2−j−N −3 · max{||s0 ||, ||s1 ||, ||s2 ||}
< 2N +1 α0 − i · 2−j−N −3 + α1 − k · 2−j−N −3 .
Since α0 , α1 ∈ [0, 1]R , there exist i, k ≤ 2j+N +3 , such that
α0 − i · 2−j−N −3 ≤ 2−j−N −3 ∧ α1 − k · 2−j−N −3 ≤ 2−j−N −3 .
If k ≤ 2j+N +3 − i, then we are finished, since
2N +1 (2 · 2−j−N −3 ) = 2−j−1 < 2−j .
22
Hence, suppose k > 2j+N +3 − i. Then we also must have i > 0. Note that
α0 ≥ (i − 1)2−j−N −3 ∧ α0 − (i − 1) · 2−j−N −3 ≤ 2 · 2−j−N −3 = 2−j−N −2 .
Since
α1 ≤ 1 − α0 ≤ 1 − (i − 1)2−j−N −3 = 2j+N +3 − (i − 1) 2−j−N −3 ,
there exists k 0 ≤ 2j+N +3 − (i − 1), such that
α1 − k 0 · 2−j−N −3 ≤ 2−j−N −3 .
Moreover, we have
2N +1 α0 − (i − 1) · 2−j−N −3 + α1 − k 0 · 2−j−N −3 ≤ 2N +1 2−j−N −2 + 2−j−N −3
= 2−j−1 + 2−j−2
< 2 · 2−j−1 = 2−j .
Hence, we conclude that S is indeed a compact metric space.
Definition 3.10. The following definition is made in RCA0 . Suppose S is a triangle in
R2 , determined by the vertices s0 , s1 , s2 ∈ R2 . A face of S is the convex hull of either one
of the vertices, the convex hull of one of the combinations of two different vertices, or S
itself. For any point x ∈ S, the carrier of x is the smallest face of S which contains x.
Definition 3.11. (triangular subdivision, [1] Definition IV.7.2). Within RCA0 , let S
be a triangle in R2 . A triangular subdivision of S is a finite set of (codes for) triangles
S0 , . . . , Sm such that S = S0 ∪ . . . ∪ Sm and, for all i < j ≤ m, Si ∩ Sj is either empty or
a common face of Si and Sj .
Definition 3.12. (admissible labeling, [1] Definition IV.7.3). Within RCA0 , let S be a
triangle, and let P be a finite set of points in S which includes the vertices of S. An
admissible labeling of P is a mapping from P into {0, 1, 2} such that
• the vertices of S are mapped to the full set of labels {0, 1, 2};
• for every x ∈ P , the label of x is the same as the label of one of the vertices of the
carrier of x.
Lemma 3.13. (Sperner’s lemma, [1] Lemma IV.7.4). The following is provable in RCA0 .
Let S be a triangle in R2 , and let S0 , . . . , Sm be a triangular subdivision of S. Suppose
that the vertices of S0 , . . . , Sm are admissibly labeled. Then for some i ≤ m, the vertices
of Si are mapped to the full set of labels {0, . . . , k}.
Proof. See [1] Section IV.7 for the proof of a more general version of Sperner’s lemma
for simplices in Rn . The proof consists of elementary combinatorial reasoning which is
straightforwardly formalized in RCA0 , and can be easily simplified to the two-dimensional
case. See also Figure 6 for an illustration.
23
Figure 6: Triangular subdivision of the triangle S in R2 , determined by the vertices
s0 , s1 , s2 ∈ R2 , in nine triangles. The vertices of S0 , . . . , S9 are admissibly labeled, and the
vertices of S6 are mapped to the full set of labels {0, 1, 2}.
Before we present our version of Brouwer’s fixed point theorem, we note that RCA0 is
sufficient to proof the fixed point theorem for the unit interval.
Theorem 3.14. The following is provable in RCA0 . It holds for every continuous function
f : [0, 1] → [0, 1], that there is a point x ∈ [0, 1] such that f (x) = x.
Proof. See the proof of [2] Theorem 5.1.(a). The proof is similar to the proof of the
intermediate value theorem in RCA0 , see [1] Theorem II.6.6 for more details.
We shall now begin with the proof of our version of Brouwer’s fixed point theorem.
Definition 3.15. (modulus of uniform continuity, [1] Definition IV.2.1). The following
definition is made in RCA0 . Let X and Y be complete separable metric spaces, and let
F be a continuous function from X into Y . A modulus of uniform continuity for F is a
function h : N → N such that for all n ∈ N and all x and y in X, if F (x) and F (y) are
defined and d(x, y) < 2−h(n) , then d(F (x), F (y)) < 2−n . We call a continuous function F
uniformly continuous, if F has a modulus of uniform continuity.
The proofs of the following two theorems make use of the Heine/Borel property which
essentially states that any covering by open sets of a compact metric space has a finite
subcovering. This assertion is provable in WKL0 .
24
Theorem 3.16. The following is provable in WKL0 . Let X be a compact metric space.
Let C be a closed set in X, and let F be a continuous function from C into a complete
separable metric space Y . Then F has a modulus of uniform continuity on C. If in
addition X = C and Y = R, then F attains a maximum value.
Proof. See [1] Theorem IV.2.2.
Theorem 3.17. The following is provable in WKL0 . Let f be a continuous real-valued
function on a nonempty closed set C in a compact metric space. If α = supx∈C f (x) exists,
then f (x0 ) = α for some x0 ∈ C.
Proof. Actually, it holds that this assertion is equivalent to WKL0 over RCA0 by [1]
Exercise IV.2.10.
It’s easy to see that this theorem also implies the same statement for the infimum.
This fact, Sperner’s Lemma, the property that a triangle can be seen as a compact metric
space by Theorem 3.9, the definition of the diameter of a triangle and the definition of a
triangular subdivision, can be used for the proof of the following lemma, stating a version
of Brouwer’s fixed point theorem for a triangle in R2 .
Lemma 3.18. ([1] Lemma IV.7.5) The following is provable in WKL0 . Let S be a triangle
in R2 . Then every continuous function f : S → S has a fixed point, i.e., f (x) = x for some
x ∈ S.
Proof. Again, see [1] Section IV.7 for the proof that any continuous function from a general
simplex in Rn to itself has a fixed point. The proof for the two-dimensional case goes
exactly the same.
By now, we are ready to show that weak König’s lemma implies Brouwer’s fixed point
theorem for the unit square. The proof is based on the proof of the general version of
Brouwer’s fixed point theorem in [1] Section IV.7.
Theorem 3.19. (Brouwer’s fixed point theorem for the unit square). The following is
provable in WKL0 . For any continuous function f : [0, 1]2 → [0, 1]2 , there is a point
x ∈ [0, 1]2 such that f (x) = x.
Proof. If we can find a triangle S in R2 such that [0, 1]2 ⊆ S, and that [0, 1]2 is a retract of
S, i.e., there is a continuous function r : S → [0, 1]2 such that r(x) = x for all x ∈ [0, 1]2 ,
then we are done. Since, given a continuous function f : [0, 1]2 → [0, 1]2 , we can consider
g : S → [0, 1]2 given by g(x) = f (r(x)). By Lemma 3.18, let x ∈ S be such that g(x) = x.
Then x ∈ [0, 1]2 , hence r(x) = x, hence f (x) = f (r(x)) = g(x) = x, i.e., x is a fixed point
for f .
Consider the triangle S defined by the vertices (0, 0), (0, 2), (2, 0) ∈ R2 . It holds that
[0, 1]2 ⊆ S and (1, 1) ∈ ∂S, see also Figure 7. Now we can easily define a continuous
function r with the right properties. Let for (a, b) ∈ S, r((a, b)) = (a, b) if (a, b) ∈ [0, 1]2 ,
r((a, b)) = (a, 1) if b > 1 and r((a, b)) = (1, b) if a > 1. (We will not perform the coding of
this function in this proof. In Subsection 4.2.3, we shall prove the existence of a code for
quite some of such ‘metric projections’, and one should be sure that this is possible in
RCA0 .)
25
Figure 7: The unit square in a 2-simplex.
This theorem can be generalized as follows.
Theorem 3.20. (Brouwer’s fixed point theorem in WKL0 ). The following is provable in
WKL0 . Let C be the convex hull of a nonempty finite set of points in Rn , n ∈ N. Then
every continuous function f : C → C has a fixed point.
Proof. The proof is essentially the same as in the proof of Theorem 3.19. However, it is a
lot harder to find a right retraction in the general case. One has to make extensive use of
linear algebra which can be developed within RCA0 . We refer the interested reader to [1]
Theorem IV.7.6. Also, in [2] Theorem 5.3 a more detailed proof is presented for the case
where the vertices are points in Qn , explicitly demonstrating how one makes use of linear
algebra.
We shall now obtain a reversal showing that weak König’s lemma is needed to prove
Brouwer’s theorem for the unit square. The proof comes directly from [1] Section IV.7, but
with added details on the singular covering of the rational interval [0, 1]. The proof is as an
example how one explicitly shows the equivalence of an ordinary mathematical statement
to WKL0 over RCA0 . Note that the following theorem also shows that the statement
given in Theorem 3.20 implies WKL0 over RCA0 . This is essentially the method Simpson
uses to prove that the general version of Brouwer’s fixed point theorem is equivalent to
WKL0 over RCA0 .
Theorem 3.21. ([1] Theorem IV.7.7). The following are pairwise equivalent over RCA0 .
26
1. Weak König’s lemma.
2. For any continuous function f : [0, 1]2 → [0, 1]2 , there is a point x ∈ [0, 1]2 such that
f (x) = x.
Proof. We need to prove 2. → 1., the other direction is proved by Theorem 3.19. Assume
that 1. does not hold. Then there exists an infinite tree T ⊆ 2<N with no infinite path.
We will now use T to construct a retraction f of C = [0, 1]2 onto the four sides ∂C
of this square. If such an f is constructed, 2. does not hold. For if f is the rotation of
90◦ about the point (0.5, 0.5), r ◦ f is a continuous function from [0, 1]2 into itself which
has no fixed point. Moreover, we will show that T implies the existence of a singular
covering of [0, 1], i.e., a covering of [0, 1] by an infinite sequence of closed rational intervals
In = [an , bn ], an , bn ∈ Q, an < bn , n ∈ N, such that for all m 6= n, Im ∩ In consists of at
most one point. This singular covering will allow us to construct the desired function f .
Define rational intervals [cσ , dσ ], σ ∈ 2<N , by putting chi = 0, dhi = 1, cσ_h0i = cσ ,
cσ_h1i = dσ_h0i = (cσ + dσ )/2, and dσ_h1i = dσ , see Figure 8. Let hσn : n ∈ Ni be an
enumeration8 of the set
Te = {σ ∈ 2<N : σ ∈
/ T ∧ σ [lh(σ) − 1] ∈ T } ,
which clearly exists in RCA0 , and put
In := [an , bn ] := [cσn , dσn ] = {q ∈ Q : cσn ≤ q ≤ dσn } .
We will show that for m =
6 n, Im ∩ In consists of at most one point. It is clear that for
m 6= n, if Im ∩ In consists of more than one point, then one is a sub-interval of the other.
W.l.o.g. suppose m 6= n and Im ⊆ In . It should be clear that this implies that σn is an
initial segment of σm [lh(σm ) − 1]. But σm ∈ Te, hence σm [lh(σm ) − 1] ∈ T and T is closed
under initial segments, hence σn ∈ T, which contradicts σn ∈ Te. So for m =
6 n, Im and In
have at most one point in common.
Now take any real number x between 0 and 1. We need to prove that x is contained in
x
one of the intervals In , n ∈ N. For
in 2<N of
k ∈ N, we let σk be the leftmost sequence
length k, such that x ∈ cσkx , dσkx . Then one easily sees that for k ≤ k 0 , σkx is an initial
segment of σkx0 . Defining,
g : N → N : k 7→ σkx (k) ,
g defines a path through 2<N . Hence, σkx must be contained in Te for some k ∈ N, otherwise
we would have a path through T , a contradiction.
Hence, we conclude that hIn : n ∈ Ni has the desired properties of a singular covering
of [0, 1].
Now for each n ∈ N we put
!
!
[
[
An =
Im × In ∪
In × Im
m≤n
We note that C =
8
S
n∈N
m≤n
An .
The existence of this enumeration is provable in RCA0 , cf. [1] Section II.3.
27
Our retraction map f : C → ∂C will be defined in stages. We begin by defining f on
∂C to be the identity map. At stage n, we assume that f has already been defined on
∂C and on Am for all m < n, and we define f on An . Let Pn0 , . . . , Pnkn be the connected
components of An . We can easily
S observe that each Pni has at least one free side eni
on which Pni does not intersect m<n Am or any side of ∂C, except for its endpoints,
see Figure 9. Let e0ni be eni minus its endpoints. We define f on Pni to consist of a
continuous retraction of Pni on ∂Pni \e0ni , followed by a continuous mapping of ∂Pni \e0i into
∂C whichSis compatible with the part of f that has already been defined. This defines f
on An = i≤kn Pni . It can be shown
S that9 the above construction gives rise to a continuous
function f defined on all C = n∈N An . Clearly f is a retraction of C onto ∂C, which
completes the proof.
Figure 8: Partitioning the unit interval using the infinite tree 2<N in RCA0 .
Figure 9: Each P2i , i = 0, . . . , 4, of the connected components of A2 has at least one free
side e2i on which it does not intersect A0 ∪ A1 or any side of ∂C, except for its endpoints.
9
We will ommit the details on the code of this function.
28
4
4.1
Browder-Göhde-Kirk’s fixed point theorem
The statement in Z2
In this chapter, we shall perform the reverse mathematics of the Browder-Göhde-Kirk
fixed point theorem, i.e., we shall look to which of the five natural subsystems of Z2
this theorem is equivalent too, over RCA0 or possibly one of the other subsystems. The
ordinary mathematical statement of the Browder-Göhde-Kirk fixed point theorem is as
follows.
Theorem 4.1. (Browder-Göhde-Kirk). Let E be a uniformly convex Banach space, let
K ⊆ E be nonempty, bounded, closed, and convex, and let f : K → K be nonexpansive.
Then f has a fixed point.
Given our purpose, we of course need to interpret this theorem in the context of the
subsystems of Z2 , working with our defined concepts. As in the case with the definition
of a uniformly continuous function (Definition 3.15), we will not consider the ordinary
mathematical -δ-definition for a uniformly convex Banach space, but rather define a
Banach space to be uniformly convex when it has a modulus of uniform convexity.
Definition 4.2. (modulus of uniform convexity). The following definition is made in
b be a separable Banach space. A modulus of uniform convexity for E
b is a
RCA0 . Let E
b
function h : N → N, such that for all n ∈ N and for all x, y ∈ E with ||x||, ||y|| ≤ 1, if
b has a modulus of uniform convexity,
||x − y|| ≥ 21−n , then ||(x + y)/2|| ≤ 1 − 2−h(n) . If E
b a uniformly convex Banach space.
we call E
Definition 4.3. (nonexpansive function). The following definition is made in RCA0 . Let
Φ denote a code for a continuous function φ between two complete separable metric spaces
b and E.
b We call φ nonexpansive if
A
d(φ(x), φ(y) ≤ d(x, y) ,
for all x, y ∈ dom(φ).
b as defined in Definition 2.12. In
We will not work with general closed sets K in E
b Having positive information in stead
stead, we will assume K to be separably closed in E.
of negative information shall make it possible to easily restrict to the relative topology
b induces on K (Theorem 4.48).
that E
Definition 4.4. (separably closed set). The following definition is made in RCA0 . Suppose
b is a complete separable metric space. A code for a separably closed set K in E
b is defined
E
b We define a point in K to be a point x in E,
b
to be a sequence hxn : n ∈ Ni of points in E.
such that
∀ ∈ R+ (∃n(d(xn , x) < )) .
b denote
Definition 4.5. (bounded set). The following definition is made in RCA0 . Let E
b
a complete separable metric space, and K an open or (separably) closed subset of E,
b if there exists
following Definition 2.12 or Definition 4.4. We say that K is bounded in E,
29
b and a constant r > 0,10 such that d(x, y) < r for all y ∈ K, i.e., if K is contained
x∈E
in a ball of finite radius.
Since ||y|| ≤ ||x|| + ||y − x||, it is easily seen that for K a subset of the separable
b K is bounded iff there exists M > 0 such that ||y|| ≤ M for all y ∈ K.
Banach space E,
b denote a
Definition 4.6. (convex set). The following definition is made in RCA0 . Let E
b We
complete separable metric space, and K an open or (separably) closed subset of E.
b if
say that K is convex in E,
x, y ∈ K ∧ 0 ≤ t ≤ 1 → (1 − t)x + ty ∈ K .
Hence, we can state Browder-Göhde-Kirk’s theorem completely using defined concepts
within RCA0 :
b
b be a uniformly convex separable Banach space, let K ⊆ E
Theorem 4.7. Let E
be nonempty, bounded, separably closed, and convex, and let Φ be a code for a
continuous nonexpansive function f : K → K. Then f has a fixed point.
The purpose of this chapter and this thesis is to perform the reverse mathematics of
this statement, also denoted by BGK.
4.2
b a Hilbert space and K a
The reverse mathematics for E
closed ball
b is given by a specific type of
In this section, we shall take a look at the case where E
separable Banach space, namely a Hilbert space.
Definition 4.8. (separable Hilbert spaces). Within RCA0 , we define a (code for a)
b to consist of a countable vector space E over the rational field
separable Hilbert space E
Q together with an inner product, i.e., a sequence of real numbers h·, ·i : A × A → R
satisfying
1. ha1 , a2 i = ha2 , a1 i ;
2. hqa1 + pa2 , bi = qha1 , bi + pha2 , bi ;
3. hb, bi ≥ 0 and the equality holds iff b = 0 ,
b is defined to be a sequence hak : k ∈ Ni of
for all a1 , a2 , b ∈ E and q, p ∈ Q. A point of E
elements of A such that hak − ak+1 , ak − ak+1 i ≤ 2−2(k−1) for all k ∈ N.
b is a separable Hilbert space, then the sequence
If E
p
|| · || : A → R : a 7→ ha, ai
10
When not specified, we will assume a constant to be a real number.
30
defines a norm on E as in Definition 2.19. This follows from the well-known Cauchy-Schwarz
inequality:
|ha, bi| ≤ ||a|| · ||b|| , ∀a, b ∈ E ,
where the equality holds iff a and b are linearly dependent in E, i.e., there exists some real
number λ such that a = λb. The Cauchy-Schwarz inequality can be proved within RCA0
in the usual way. Hence, under this identification, E also defines a code for a separable
b and R.
Banach space. Again, all relevant operations are within RCA0 extendable to E
Any separable Hilbert space is also a uniformly convex Banach space. This follows
from the following theorem:
b be a Hilbert space. Then E
b is
Theorem 4.9. The following is provable in RCA0 . Let E
uniformly convex.
Proof. One can check in the usual way that the parallelogram law holds for a Hilbert
b i.e., for x, y ∈ E,
b we have
Space E,
2||x||2 + 2||y||2 = ||x + y||2 + ||x − y||2 ,
which implies
2
1
1
1
1
||x + y|| = ||x||2 + ||y||2 − ||x − y||2 .
2
2
2
4
b are such that ||x||, ||y|| ≤ 1 and ||x − y|| ≥ 21−n .
Hence, suppose n ∈ N, and x, y ∈ E
Applying the parallelogram law, we find
r
x + y 22−2n √
−2n .
2 ≤ 1 − 4 = 1 − 2
Defining
h : N → N : n 7→ 4n ,
we find
21−h(n) − 2−2h(n) = 21−4n − 2−8n ≤ 2−2n ,
for all n ∈ N, so that
1 − 2−2n ≤ 1 − 21−h(n) + 2−2h(n) = 1 − 2−h(n)
2
.
b
Hence, h is a modulus of uniform convexity for E.
It is easy to see that Theorem 2.14 implies the existence of any closed ball B(a, r) with
b and r ∈ R+ , which is clearly a nonempty, bounded and convex set. We will show
a∈E
b a fixed Hilbert space and K a fixed closed ball, denoted by BGKH , is
that BGK for E
equivalent to ACA0 over RCA0 .
31
Proving that ACA0 → BGKH over RCA0
4.2.1
For the direction ACA0 → BGKH , we shall follow a strategy similar to the one applied in
[3].
b
b be a Hilbert space, u, v ∈ E
Lemma 4.10. The following is provable within RCA0 . Let E
b such that
and r, R be constants such that 0 ≤ r < R. Suppose there exists an x ∈ E
u + v
||u − x|| < R , ||v − x|| < R , − x ≥ r .
2
Then
√
||u − v|| < 2 R2 − r2 .
Proof. Writing u − v = (u − x) − (v − x), one can easily check this inequality by applying
the parallelogram law.
Lemma 4.11. The following is provable in RCA0 . Suppose K is a closed subset of the
b f : K → K is nonexpansive and that there exists a positive constant L
Hilbert space E,
such that for all x, y ∈ K, ||x − y|| ≤ L.11 Suppose that x, y, a := (x + y)/2 ∈ K and that
≤ L is such that ||x − F (x)|| < and ||y − F (y)|| < . Then
√√
||a − F (a)|| < 2 2L .
Proof. The triangle inequality for separable Banach spaces implies
a
+
F
(a)
a
+
F
(a)
+ y −
.
||x − y|| ≤ x −
2
2
Hence,
1
1
a
+
F
(a)
a
+
F
(a)
≥ ||x − y|| or y −
≥ ||x − y|| ,
x −
2
2
2
2
or the first inequality wouldn’t hold. W.l.o.g., we can hence assume that
x − a + F (a) ≥ 1 ||x − y|| .
2
2
Now, since
x + y
1
1
||a − x|| = − x = ||x + y − 2x|| = ||x − y|| ,
2
2
2
we have
1
||F (a) − x|| ≤ ||F (a) − F (x)|| + ||F (x) − x|| < ||a − x|| + = ||x − y|| + .
2
Now we put
1
1
r := ||x − y||, R := ||x − y|| + and u = a, v = F (a) .
2
2
11
This implies K is bounded and vice versa.
32
By these definitions, the assumptions of the previous lemma are fulfilled and we have
s
2 2
1
1
||x − y|| + −
||x − y||
||a − F (a)|| < 2
2
2
p
√p
≤ 2 ||x − y|| + 2 = 2 ||x − y|| + √√
≤ 2 2L .
The following result is known as Banach’s contraction principle.
Theorem 4.12. (Banach’s contraction principle). The following is provable in RCA0 .
b and f : K → K is a
Suppose K is a nonempty and closed subset of the Hilbert space E
contraction, i.e., a Lipschitz-continuous function with Lipschitz-constant 0 ≤ L < 1. Then
f has a unique fixed point.
Proof. We first prove the uniqueness. Suppose there exists x, y ∈ K such that f (x) = x
and f (y) = y, then
||x − y|| = ||f (x) − f (y)|| ≤ L||x − y|| ,
and since 0 ≤ L < 1, we must have x = y.
Now take x ∈ K. Within RCA0 , it is possible to define the sequence hf n (x) : n ∈ Ni,
where f n : K → K is the function formed by applying f n times. For n ∈ N, we have
||f n (x) − f n+1 (x)|| ≤ L||f n−1 (x) − f n (x)|| ≤ . . . ≤ Ln ||x − f (x)|| ,
so that, for m > n ∈ N, we have
||f n (x) − f m (x)|| ≤ ||f n (x) − f n+1 (x)|| + . . . + ||f m−1 (x) + f m (x)||
≤ Ln ||x − f (x)||(1 + L + . . . + Lm−n−1 )
≤
Ln
||x − f (x)|| .
1−L
b and
Hence, hf n (x) : n ∈ Ni is a Cauchy-sequence in the complete separable space E
b Since K is closed, this
Theorem 2.10 implies the convergence of this sequence in E.
sequence must converge in K by Theorem 2.13. Hence we have an u ∈ K such that,
applying the continuity of f (Theorem 2.16)
u = lim f n+1 (x) = lim f (f n (x)) = f (u) ,
n→∞
n→∞
which implies that u is a fixed point of f in K.
Theorem 4.13. (Cantor’s intersection theorem). The following is provable in RCA0 . Let
hCn : n ∈ Ni be a decreasing sequence of nonempty closed sets in a complete separable
b i.e., for all m, n ∈ N, m ≥ n implies Cm ⊆ Cn . Suppose there exists a
metric space E,
decreasing sequence of reals hn : n ∈ Ni with limn n = 0 such that u, v ∈ Cn implies
||u − v|| ≤ n . If there exists a sequence of points hxn : n ∈ N, xn ∈ Cn i, then this sequence
b such that x ∈ Cn for all n ∈ N.
converges to the unique point x ∈ E
33
Proof. The uniqueness is clear, since if both x, y ∈ Cn for all n ∈ N, then ||x − y|| ≤ n
for all n ∈ N. Now for m, n ∈ N, m ≥ n, we have
||xm − xn || ≤ n ,
b
since xm ∈ Cm ⊆ Cn . Hence, by Theorem 2.10, hxn : n ∈ Ni converges to some x ∈ E.
We claim that x ∈ Cn for all n ∈ N. Suppose not, then there exists k ∈ N such that
x∈
/ Ck . But then we must have x ∈ Uk , where Uk denotes the open complement of Ck ,
i.e., ∃(i, a, r) ∈ Uk ⊆ N × E × Q+ with d(x, a) < r. Now let n ∈ N be such that n ≥ k and
d(x, xn ) < (r − d(a, x))/2, then
r + d(x, a) r − d(x, a)
+
= r,
2
2
so that xn ∈ Uk , i.e., xn ∈
/ Ck ⊇ Cn , a contradiction. Hence, we conclude x ∈ Cn for all
n ∈ N.
d(a, xn ) ≤ d(a, x) + d(x, xn ) <
Remark 4.14. In ordinary mathematics, one doesn’t need the assumption that the
sequence hxn , n ∈ N, xn ∈ Cn i is given. One just starts by taking ‘such a’ sequence without
any further justification. The problem in the context of reverse mathematics in proving the
statement when no such sequence is given, lies in proving the existence of such a sequence.
This is possible applying weak König’s lemma when the Cn ’s are contained in a compact
metric space, see [1] Theorem IV.1.8. However, in general Hilbert spaces, the existence of
such a sequence is probably too complex for ACA0 to prove. Note that even in ordinary
mathematics, the properties bounded and closed are not always equivalent to compactness.
As we shall see, it is a lot easier to extract a sequence of points hxn , n ∈ N, xn ∈ Un i, when
hUn : n ∈ Ni denotes a sequence of nonempty, open sets in a complete separable metric
space.
b be a separale Banach space,
Lemma 4.15. The following is provable in ACA0 . Let E
b and g : K → R continuous. Then the set
K = BEb (u, s) a closed ball in E
Z = (a, q, r) ∈ E × Q+ × Q+ : ∃x ∈ BEb (a, q) (g(x) < r)
exists.
Proof. Let Φ ⊆ N × E × Q+ × Q × Q+ be the code for g. We will show that the property
ϕ ≡ ∃x ∈ BEb (a, q) (g(x) < r)
is equivalent the arithmetical formula:
ψ ≡ ∃(n, a0 , q 0 , b, r0 ) ∈ N × E × Q+ × Q × Q+

ψ1
z
}|
{

0 0
0 0
0
((n, a , q , b, r ) ∈ Φ) ∧ BEb (a , q ) ∩ K 6= ∅


∧ BEb (a0 , q 0 ) ⊆ BEb (a, q) ∧ BR (b, r0 ) ⊆ ] − ∞, r[ 
,
{z
} |
|
{z
}
ψ2
34
ψ3
which implies the existence of Z in ACA0 . Note that this formula is indeed arithmetical,
moreover, we have
ψ1 ↔ ||a0 − u|| − s < q 0 ;
ψ2 ↔ ||a − a0 || + q 0 ≤ q ;
ψ3 ↔ b + r0 < r .
First suppose ψ. We want to show that ϕ holds, i.e., we have g(x) < r for some
x ∈ BEb (a, q). Let (n, a0 , q 0 , b, r0 )) ∈ Φ be such that ψ1 ∧ ψ2 ∧ ψ3 holds. By ψ1 , there exists
x ∈ BEb (a0 , q 0 ) ∩ K. Then g(x) is defined, and by ψ2 , we have x ∈ BEb (a, q). Moreover,
since (n, a0 , q 0 , b, r0 ) ∈ Φ implies g(BEb (a0 , q 0 ) ∩ K) ⊆ BR (b, r0 ), we have g(x) < r by ψ3 .
Now suppose ϕ, then we have x ∈ BEb (a, q) such that g(x) < r. Since x ∈ K = dom(g),
there exists (n1 , a01 , q10 , b, r0 ) ∈ Φ with ||x − a01 || < q10 and |g(x) − b| ≤ r0 < (r − g(x))/2.
We then have
r0 = r+r0 −
r − g(x) r − g(x)
r − g(x) r − g(x)
−
−g(x) < r+
−
−(b−g(x))−g(x) = r−b ,
2
2
2
2
from which we conclude BR (b, r0 ) ⊆ ] − ∞, r[. Now let a0 ∈ E be such that
0
q1 − ||x − a01 || q − ||x − a||
0
,
||x − a || < min
=: q 0 ,
2
2
then we have
||a01 − a0 || + q 0 ≤ ||a01 − x|| + ||x − a0 || + q 0 < ||a01 − x|| +
q10 − ||x − a01 || q10 − ||x − a01 ||
+
= q10 .
2
2
Hence, there exists (n, a0 , q 0 , b, r0 ) ∈ Φ by Property 2 of continuous functions, given in
Definition 2.15. Note that x ∈ BEb (a0 , q 0 ) ∩ K. Moreover, we have
q 0 ≤ q − ||x − a|| − q 0 < q − ||x − a|| − ||x − a0 || ≤ q − ||a − a0 || ,
where the first inequality follows from the definition of q 0 . Hence, BEb (a0 , q 0 ) ⊆ BEb (a, q),
from which we conclude that ϕ implies ψ, which ends our proof.
Remark 4.16. Note that it is important to restrict ourselves to separable Banach spaces
b is just a complete separable metric space, equivalences such as
in the previous lemma. If E
B(u, s) ∩ B(v, t) 6= ∅ ↔ ||u − v|| < s + t ,
b s, t ∈ R+ do not necessarily hold. For suppose the complete separable space
with u, v ∈ E,
b is the union of the two real intervals
E
b = [−3, −1] ∪ [1, 3] .
E
b Then
Now consider the open ball B(−2, 2) and closed ball B(2, 5/2) in E.
|2 − (−2)| = 4 <
35
9
5
=2+ ,
2
2
but
B(−2, 2) = [−3, −1] ,
B(2, 5/2) = [1, 3] ,
b would be a separable Banach space, then we know
so that B(−2, 2) ∩ B(2, 5/2) = ∅. If E
b and we have −1/3 ∈ B(−2, 2) ∩ B(2, 5/2).
−1/3 ∈ E,
b is a complete separable
Lemma 4.17. The following is provable in RCA0 . Suppose E
metric space, and hUn : n ∈ Ni and hVn : n ∈ Ni are both sequences of (codes for) open
b Then there exists a sequence of (codes for) open sets hWn : n ∈ Ni in E,
b such
sets in E.
that Wn = Un ∩ Vn for all n ∈ N.
Proof. Note that this Lemma is a variant of [1] Exercise II.5.11, which states that given a
finite sequence of (codes for) open sets, it’s possible to find (a code for) an open set which
denotes the intersection of all these sets within RCA0 .
b and n ∈ N, we define the Σ0 formula
For x ∈ E
1
ϕ(n, x) ≡ ∃(l, a, r) ∈ Un (d(x, a) < r) ∧ ∃(k, b, s) ∈ Vn (d(x, b) < s) .
b such
By Theorem 2.14, there exists a sequence of (codes for) open sets hWn : n ∈ Ni in E,
that x ∈ Wn ↔ ϕ(n, x). It is clear that Wn = Un ∩ Vn for all n ∈ N.
In our proof of ACA0 →BGKH , we shall need to restrict to the relative topology that
b
E induces on K. This can be done by the following lemma:
b be a separable Banach space,
Lemma 4.18. The following is provable in ACA0 . Let E
b Then B(a, r) is a complete separable metric
b r > 0 and B(a, r) a closed ball in E.
a ∈ E,
space.
b and || · || the norm on E.
b By arithmetical comprehension,
Proof. Let E be the code for E
we define
L = {x ∈ E : ||a − x|| < r} ,
and the sequence of reals
dm : L × L → R : (x, y) 7→ ||x − y|| ,
i.e., dm is the restriction of the metric on E to L. Then we have a code for the complete
b It should be clear that any point in L is a point of K. Since
separable metric space L.
b
a ∈ E, we have a = hak : k ∈ N, ak ∈ Ei with ∀i ∀j(i < j → ||ai − aj || ≤ 2−i ). Since
limk ||a − ak || = 0, we must have n ∈ N such that m ≥ n implies ||a − am || < r. Defining
a0k = ak+n , it is easily seen that a = ha0k : k ∈ N, a0k ∈ Li defines a sequence in L, with
b
∀i ∀j(i < j → dm (a0i , a0j ) ≤ 2−(i+n) ≤ 2−i ), so that a ∈ L.
Now let x = hxk : k ∈ N, xk ∈ Li be such that ∀i ∀j(i < j → dm (xi , xj ) ≤ 2−i ). Since
dm (a, xk ) < r for every k ∈ N, we have
dm (a, x) = lim dm (a, xk ) ≤ r ,
k
b ⊆ K.
so that x ∈ K and L
36
Now suppose x ∈ K, i.e., ||a − x|| ≤ r, where x = hxk : k ∈ N, xk ∈ Ei is such that
∀i ∀j(i < j → ||xi − xj || ≤ 2−i ). If ||a − x|| = limk ||a − xk || < r, then there must be
n ∈ N such that m ≥ n implies ||a − xm || < r. Again, this allows us to convert the given
sequence for x into an appropriate sequence of elements in L for x. Hence, suppose that
b essentially follows from
x lies on the boundary of B(a, r), i.e., ||a − x|| = r. Then x ∈ L
b By this, we conclude that L with the metric dm
the proof of Lemma B.1. Hence, K ⊆ L.
b = K.
determines a code for the complete separable metric space L
Remark 4.19. By [1] Lemma II.3.7, there exists an enumeration of the (informal) set
L = {x ∈ E : ||a − x|| < r} ,
b
within RCA0 , which shows that the closed ball B(a, r) is a separably closed set in E.
Theorem 4.20. ACA0 implies BGKH over RCA0 .
b be a Hilbert space, K a closed ball in E
b and f : K → K nonexpansive.
Proof. Let E
Since K is nonempty, we may take x0 ∈ K.
Now assume f (x0 ) 6= x0 , otherwise we would be finished. For each n ∈ N with n ≥ 2,
we define
1
1
fn := x0 + 1 −
f :K →K,
n
n
note that indeed f maps K to K since K is convex. Moreover, we have for x, y ∈ K
1
1
||fn (x) − fn (y)|| = 1 −
||f (x) − f (y)|| ≤ 1 −
||x − y|| ,
n
n
which implies fn is a contraction. Hence, by Banach’s contraction principle, fn has a
unique fixed point xn ∈ K with
1
1
xn = fn (xn ) = x0 + 1 −
f (xn ) .
n
n
Now, since K is bounded, there exists M > 0 such that
||xn − f (xn )|| =
1
2M
||x0 − f (xn )|| <
.
n
n
By the previous lemma, it is justified working within K as if K is a complete separable
metric space itself. We want to find a sequence of codes for open sets hQn : n ∈ N, n ≥ 2i,
where Qn denotes the informal open set
2M
Qn = x ∈ K : ||x − f (x)|| <
,
n
b we define the Σ01 -formula
with n ∈ N and n ≥ 2. For n ∈ N and x = hak : k ∈ Ni ∈ E,
ϕ(x, n) ≡ ||x − f (x)|| <
37
2M
.
n
Now suppose x and y represent the same points in K, i.e., ||x − y|| = 0 and suppose
ϕ(x, n), i.e.,
2M
||x − f (x)|| <
.
n
b we have the
Since we know f (x) to be defined uniquely up to equality of points in E,
equality f (x) = f (y) in K, so that
||(x − f (x)) − (y − f (y))|| ≤ ||x − y|| + ||f (x) − f (y)|| = 0 .
Hence, ||(x − f (x)) − (y − f (y))|| = 0, from which
||y − f (y)|| ≤ ||x − f (x)|| + ||(x − f (x)) − (y − f (y))|| <
2M
,
n
so that we may conclude ϕ(y, n). Hence, the sequence of codes for the open sets Qn , as
subsets of the complete separable metric space K, can be defined within RCA0 by Theorem
2.14. So for every n ∈ N, n ≥ 2, we have a code for the open set Qn . It should be clear
that
Q2 ⊇ Q3 ⊇ . . . ⊇ Qn ⊇ . . . ,
is a decreasing sequence of nonempty (since xn ∈ Qn ) open sets. By Lemma 4.15, we can
define
Z = {(a, q, r) ∈ E × Q+ × Q+ : (∃x ∈ B(a, q))(||x − f (x)|| < r)} .
Note that the identity function, subtraction, the norm and the composition of continuous
function can be shown to be continuous within RCA0 , see also [1] Section II.6. Within
ACA0 , given n ∈ N, n ≥ 2, it is possible to define12
2M
+
∈Z
dn := inf q ∈ Q : 0, q,
n
the concerning set is indeed nonempty (since Qn is nonempty) and bounded from below
by 0. We claim that
dn = inf{||x|| : x ∈ Qn } ,
i.e.,
1. x ∈ Qn → ||x|| ≥ dn ;
2. (∀ > 0)(∃x ∈ Qn )(||x|| ≤ dn + ) .
Suppose x ∈ Qn , i.e., x ∈ K with ||x − f (x)|| < 2M/n. If ||x|| < dn , then there exists
q ∈ Q+ such that ||x|| < q < dn , but this implies
2M
0, q,
∈ Z ∧ q < dn ,
n
12
See also [1] Section III.2.
38
a contradiction, so that ||x|| ≥ dn . Now take any > 0. By definition of the infimum,
there must be a q ∈ Q+ such that

 0, q, 2M ∈ Z ;
n
d ≤ q ≤ d + ,
n
n
and the second property follows from the definition of Z and Qn . Hence, dn is indeed the
required infimum.
Since, the Qn ’s are decreasing, considering inclusion as the ordering, we obviously have
d2 ≤ d3 ≤ . . . ≤ dn ≤ . . . , with di ≤ M for all i ≥ 2 .
Hence, ACA0 implies the existence of
d = sup{dn : n ∈ N} ,
and we have limn dn = d ≤ M . Working in K, by Lemma 4.17 there exists a sequence of
open sets hAn : n ∈ N, n ≥ 2i, where An is a code for the open set
1
An = Q8n2 ∩ B 0, d +
⊆K,
n
and B(0, d + 1/n) can be definied as an open subset of K by Theorem 2.14. Note that
hAn : n ∈ N, n ≥ 2i defines a decreasing sequence of nested, open sets and that by
definition of the infimum, each An is nonempty. Now for u, v ∈ An , we have
||u − 0|| = ||u|| < d +
1
1
and ||v − 0|| = ||v|| < d + .
n
n
(1)
Also, since u, v ∈ Q8n2 ,
||u − f (u)|| <
M
M
and ||v − f (v)|| < 2 .
2
4n
4n
Now Lemma 4.11, where u, v are points in the convex set K and hence (u + v)/2 ∈ K,
implies
r
√
u + v
u
+
v
< 2 M 2 · 2M = 2M ,
−
f
2
2
4n2
n
from which (u + v)/2 ∈ Qn and
u + v
u + v 2 − 0 = 2 ≥ dn .
so that Lemma 4.10, together with (1) and (2), implies
s
2
1
d+
− d2n .
||u − v|| < 2
n
39
(2)
Hence, for all u, v ∈ An , we find
r
||u − v|| < n := 2 (d2 − d2n ) +
2d
1
+ 2,
n
n
with limn n = 0.
We will show that this implies the existence of a point y contained in all open sets An ,
and hence,
M
||y − f (y)|| < 2 for all n = 2, 3, . . . ,
4n
so that y will be a fixed point of f .
Let n ∈ N, n ≥ 2. Since An is an open set in K, it is determined by some code
b = K. We
An ⊆ N × L × Q , where L is a code for the complete separable metric space L
define
an := min l ∈ L : (∃m ∈ N , ∃q ∈ Q+ )((m, l, q) ∈ An ) ,
N
then han : n ∈ N, n ≥ 2i denotes a sequence of points an ∈ An . Working within K, it is
again possible to define a sequence hUn : n ∈ N, n ≥ 3i of codes for the open sets
2M
Un = x ∈ K : ||x − f (x)|| >
,
n
D
E
en : n ∈ N, n ≥ 3 , with
which denotes a sequence of codes for closeds sets Q
en =
Q
2M
x ∈ K : ||x − f (x)|| ≤
n
the closed complement of Un in K. By Theorem 2.14, there exists a code for the open set
C
1
B (0, d + 1/n) = x ∈ K : ||x|| > d +
.
n
Hence, we can define a code for the open set
n
o
C
C
U8n2 ∪B(0,d +1/n) = (m, l, q) ∈ N×L×Q : (m, l, q) ∈ Un ∨(m, l, q) ∈ B (0, d + 1/n) ,
which denotes the open complement in K of the closed set
en = Q
e8n2 ∩ B(0, d + 1/n) .
A
en , n ∈ N, n ≥ 3i denotes a sequence of closed sets within K. Note that for
Now hA
en and x ∈ A
en implies x ∈ An−1 . Hence, applying Cantor’s
n ≥ 3, an ∈ An implies an ∈ A
intersection theorem, we know that han : n ∈ N, n ≥ 2i converges to a (unique) point
en for n ≥ 3 and hence within all An for n ≥ 2, which completes the
contained within all A
proof.
40
Remark 4.21. In the start of this section, we stated that our strategy in proving
ACA0 →BGKH would be similar to the approach in [3]. There is, however, a major
difference when formalizing this approach in the context of reverse mathematics.
In [3], the BGK statement is proved for arbitrary closed, bounded and convex sets K
in a Hilbert space. The observant reader should have noticed that the most important
step in our analysis which used the fact that K was a closed ball, occured in the proof
of Lemma 4.15. More specifically, the property that the intersection of a given ball with
K is nonempty was easily expressed by the arithmetical formula ψ1 , i.e., ACA0 proves
the existence of the set of all (codes for) open balls that intersect with K. If K is an
arbitrary nonempty, closed and convex set in a non-compact space, it’s possible ACA0
cannot prove the existence of such a set. Lemma 4.15 was applied to show the existence
of inf{||x|| : x ∈ Qn } in the proof of Theorem 4.20. Within ordinary mathematics, the
existence of this infimum is not that big of a deal. Within reverse mathematics, ACA0
proves the existence of this infimum if Qn is coded as an open set in a separable Banach
space. This is stated in the following theorem.
b be a separable Banach space
Theorem 4.22. The following is provable in ACA0 . Let E
b Then
and U an open set in E.
inf{||x|| : x ∈ U }
exists.
Proof. Suppose there exists (n, a, r) ∈ U such that ||a|| − r < 0. Then ||a − 0|| = ||a|| < r,
so that 0 ∈ U and inf{||x|| : x ∈ U } = 0. Now suppose that for all (n, a, r) ∈ U , we have
||a|| − r ≥ 0. By arithmetical comprehension, we define
W = {q ∈ Q : (∃(n, a, r) ∈ U )(q > ||a|| − r)} ,
and claim that
d := inf W = inf{||x|| : x ∈ U } .
Let x ∈ U , then there exists (n, a, r) ∈ U such that ||x − a|| < r, so that ||a|| − r < ||x||.
Since we can find q ∈ Q with ||a|| − r < q < ||x||, we have ||x|| > d. Now suppose
> 0. By definition of the infimum, we find q ∈ W such that q < d + , i.e., there exists
(n, a, r) ∈ U with d + > q > ||a|| − r. Since ||a|| ≥ r > 0, we can define
x :=
qa
.
||a||
W.l.o.g., we may assume q < ||a||, from which we find
qa (||a|| − q)||a||
||a − x|| = a −
=
= ||a|| − q < r ,
||a|| ||a||
so that x ∈ U . Since ||x|| = q < d + , we find that d is indeed the required infimum.
b inf{||x|| : x ∈ U } exists in
Hence, if U is coded as an open set in a Banach space E,
ACA0 . This proof does not necessarily work for the case where U is coded as an open set
b induces on a closed set K. This is illustrated in Figure 10,
in the relative topology that E
and the reason why we needed Lemma 4.15.
41
b = R2 .
Figure 10: The set B(a, q)∩K can be considered an open ball in the closed set K ⊆ E
However, we see that ||a|| − q is far less than inf{||z|| : z ∈ K}.
4.2.2
The Hilbert space l2
Within RCA0 , we define |A| ⊆ N to be the set of (codes for) nonempty finite sequences
of rational numbers hr0 , . . . , rm i such that either m = 0 or rm =
6 0. Addition on |A| is
defined by putting hr0 , . . . , rm i + hs0 , . . . , sn i = hr0 + s0 , . . . , rk + sk i where ri , si = 0 for
i > m, n respectively, and k = max{i : i = 0 ∨ ri + si 6= 0}. For scalar multiplication on
|A|, we put q · hr0 , . . . , rm i = h0i if q = 0, hq · r0 , . . . , q · rm i if 0 6= q ∈ Q. Then A becomes
a vector space over Q as in Definition 2.18.
Definition 4.23. (The Hilbert space l2 ). Let A be as above. For all hr0 , . . . , rm i ∈ |A|
we put
!1/2
m
X
||hr0 , . . . , rm i|| =
|ri |2
.
i=0
b and we define l2 = A.
b
Then A becomes a code for a separable Banach space A
Remark 4.24. l2 is indeed a Hilbert space. Moreover, the norm on l2 is defineable by
the inner product
min{lh(x),lh(y)}−1
X
h·, ·i : A × A : hx, yi 7→
xi · y i .
i=0
One may wonder what l2 , as the completion of A, looks like. It is certainly not the
informal set of all finite sequences of reals, for take a look at the sequence of finite sequences
1
1
1 1
x=
: k ≤ n : n ∈ N = h1i , 1,
, 1, ,
,... .
2k
2
2 4
42
We have for n, m ∈ N, n > m
||xn − xm || ≤ ||xn − xn−1 || + . . . ||xm+1 − xm || =
1
1
1
+ . . . m+1 ≤ m ,
n
2
2
2
b Now suppose y is a point of the separable Banach space Rk , i.e.,
so that x ∈ A.
y = hr0 , . . . , rk−1 i = hhqn0 , . . . , qnk−1 i : n ∈ N, qni ∈ Q for i ≤ k − 1i ,
with ||hqn0 , . . . , qnk i − hqm0 , . . . , qmk i|| ≤ 2−m for all m, n ∈ N with n > m. Since the
norm on Qk and the norm on A coincide for finite sequences of length k, even with zero
b Now, it is easy to see that in A,
b we have
end-terms, we also have y ∈ A.
1
1
||x − y|| = lim m : m ≤ n − hqn0 , . . . , qnk i ≥ k+1 ,
n
2
2
b is not represented by any finite sequence of reals.
so that x ∈ A
b is a separable Banach space and let hxi : i ∈ Ni denote a
Definition 4.25. Suppose E
b We say that P∞ xi converges if there exists x ∈ E
b such that
sequence of points in E.
i=0
x = lim
n
n
X
xi .
i=0
P
b When such
Note that each finite summation ni=0 xi , n ∈ N, indeed defines a point of E.
b exists, we also define
a point x ∈ E
∞
X
xi := x .
i=0
So what does our completion of A looks like? As one could already suspect, our
definition of l2 can be identified with the designation of l2 in ordinary mathematics. This
follows from the following theorem:
Theorem 4.26. The following is provable in RCA0 . The points of l2 are in P
canonical
2
one-to-one correspondence with the sequences hxi : i ∈ N, xi ∈ Ri, such that ∞
i=0 |xi |
converges. This correspondence is given by
n→∞
hhx0 , . . . , xn i : n ∈ N, xi ∈ R, i = 1, . . . , ni −−−→ x ←→
l2
Proof. Let hxn : n ∈ Ni be a sequence of reals such that
the sequence han : n ∈ Ni of points in l2 , with
an = hx0 , . . . , xn i .
43
∞
X
|xi |2 converges .
i=0
P∞
n=0
|xn |2 converges. We define
Now for n, m ∈ N, n > m, we find
p
||an − am || = |xm+1 |2 + . . . + |xn |2
!1/2
n
m
X
X
=
|xi |2 −
|xi |2
≤
i=0
i=0
∞
X
|xi |2 −
i=0
m
X
!1/2
|xi |
m→∞
−−−→ 0 .
i=0
By Theorem 2.10, the sequence han : n ∈ Ni converges to a point a ∈ l2 , the representation
in l2 of hxn : n ∈ Ni. This representation is well-defined, since it is easy to see that
||hxn : n ∈ Ni − hyn : n ∈ Ni|| = 0 iff xn = yn , ∀n ∈ N.
Conversely, suppose the sequence
hhx0 , . . . , xn i : n ∈ N, xn ∈ Ri
converges to the point a ∈ l2 . Then for the sequence hxn : n ∈ Ni, we have for all
n, m ∈ N, n > m,
n
X
2
|xi | −
i=0
m
X
|xi |2 = |xm+1 |2 + . . . + |xn |2
i=0
≤ lim ||hx0 , . . . , xn i − hx0 , . . . , xm i||2
n→∞
m→∞
= ||a − hx0 , . . . , xm i||2 −−−→ 0 ,
P
2
so that ∞
n=0 |xn | converges by Theorem 2.10. Again, it is easy to see that we have
limn ||hx0 . . . , xn i − hy0 , . . . , yn i|| = 0 in l2 iff xn = yn for all n ∈ N.
It is left to show that any point of l2 can be identified as the limit point of a converging
sequence of the form hhx0 , . . . , xn i : n ∈ N, xn ∈ Ri. Let a = han : n ∈ N, an ∈ Ai ∈ l2 .
Then a is of the form
a = qn0 , . . . , qn(lh(an )−1) : n ∈ N .
We define the sequence hxn : n ∈ Ni = hhpnk : k ∈ Ni : n ∈ Ni of real numbers such that

= qkn
if lh(ak ) > n ,
pnk
= 0
if lh(ak ) ≤ n .
Note that each xn , n ∈ N, is indeed a real number, since for k, l ∈ N, k < l, we have
|pnl − pnk | ≤ ||al − ak || ≤ 2−k .
We claim that a = limn hhx0 , . . . , xn i : n ∈ Ni, i.e.,
lim qn0 , . . . , qn(lh(an )−1) − hx0 , . . . , xn i = 0 .
n
44
We have
qn0 , . . . , qn(lh(an )−1) − hx0 , . . . , xn i2

|p0n − x0 |2 + . . . + |pnn − xn |2
=
2
2
|p − x |2 + . . . + |p − x |2 + |q
0n
0
nn
n
n(n+1) | + . . . + |qn(lh(an )−1) |

(n + 1) · 2−2n
≤
2
2
(n + 1) · 2−2n + |q
n(n+1) | + . . . + |qn(lh(an )−1) |
(if n ≥ lh(an ) − 1)
(if n < lh(an ) − 1)
(if n ≥ lh(an ) − 1)
.
(if n < lh(an ) − 1)
Clearly, we have a = limn hhx0 , . . . , xn i : n ∈ Ni if the lengths of the sequences an , n ∈ N,
are bounded. In case these lengths are not uniformly bounded, we can define
dn = min{k ∈ N : lh(ak ) > n} ,
which is well-defined, since for every n ∈ N, there must be at least one ak , k ∈ N, with
lh(ak ) > n. Now for any n, m ∈ N, m ≤ n we have
{k ∈ N : lh(ak ) > n} ⊆ {k ∈ N : lh(ak ) > m} ,
so that dm ≤ dn . Moreover, for any N ∈ N, we can take
m = max{lh(a0 ), . . . lh(aN )} .
Now if dm ≤ N , then there must exist some k ≤ N , with lh(ak ) > m ≥ lh(ak ), a
contradiction, so that dm > N . Since the lengths of the sequences an , n ∈ N, are not
uniformly bounded, we conclude that the sequence hdn : n ∈ Ni diverges monotonically to
infinity. Now suppose n is large enough so that dn − 1 ∈ N, i.e., dn > 0. If n is such that
n < lh(an ) − 1, then clearly dn − 1 < dn ≤ n. Moreover, since lh(adn −1 ) ≤ n < n + 1 by
definition of dn , we have
2−2(dn −1) ≥ ||an − adn −1 ||2
= qn0 − q(dn −1)0 , . . . , qn(lh(adn −1 )−1) − q(dn −1)(lh(adn −1 )−1) ,
2
qn(lh(adn −1 )) , . . . , qn(lh(an )−1) 2
2
≥ qn(lh(adn −1 )) + . . . + qn(lh(an )−1) 2
2
≥ qn(n+1) + . . . + qn(lh(an )−1) ,
45
so that for n large enough, we have
qn0 , . . . , qn(lh(an )−1) − hx0 , . . . , xn i2

(n + 1) · 2−2n
≤
2
2
(n + 1) · 2−2n + |q
n(n+1) | + . . . + |qn(lh(an )−1) |

(n + 1) · 2−2n
≤
(n + 1) · 2−2n + 2−2(dn −1)
(if n ≥ lh(an ) − 1)
(if n < lh(an ) − 1)
(if n ≥ lh(an ) − 1)
(if n < lh(an ) − 1)
n→∞
≤ (n + 1) · 2−2n + 2−2(dn −1) −−−→ 0 .
Hence, we again find a = limn hhx0 , . . . , xn i : n ∈ Ni.
Remark 4.27. Let hxn : n ∈ Ni be a sequence of reals such that
The ordinary l2 norm of such a sequence is then defined as
!1/2
∞
X
||hxn : n ∈ Ni|| :=
|xn |2
.
P∞
n=0
|xn |2 converges.
n=0
Clearly, the given correspondence in Theorem 4.26 is norm-preserving.
4.2.3
Proving that BGKH → ACA0 over RCA0
We shall now prove the reverse, i.e., that BGKH implies ACA0 over RCA0 , by proving a
property that’s equivalent to ACA0 over RCA0 , making use of BGKH for the unit ball
b can be identified as the completion of the
in l2 . Note that the inner product in l2 = A
sequence
min{lh(a),lh(b)}−1
X
h·, ·i : A × A → Q : (a, b) 7→
ai b i .
i=0
b
We will first need the existence in RCA0 of the projection from a Hilbert space E
b One can find a more general proof for the projection to any
to a closed ball within E.
nonempty, closed and convex subset K in a uniformly convex Banach space in [4] Theorem
2.23. However, this proof does in general not work within RCA0 .
We will start by proving an important equivalence to the Bourbaki-Cheney-Goldstein
inequality, as stated in [6], which will imply that any metric projection on a closed, convex
and nonempty set in a Hilbert space is a nonexpansive function. The general proof of the
following characterization can also be found in [7] or [8].
Lemma 4.28. The following is provable in RCA0 . Let K be a closed, convex and nonempty
b and suppose we have a continuous function P : dom(P ) → K,
subset of a Hilbert space E
b 13 Then
where K C ⊆ dom(P ) ⊆ E.
||x − P (x)|| ≤ ||x − y|| ,
13
b
K C denotes the open complement of K in E
46
∀x ∈ K C , ∀y ∈ K ,
if and only if the Bourbaki-Cheney-Goldstein inequality holds, i.e.,
hx − PK (x), z − PK (x)i ≤ 0 ,
b ∀z ∈ K ,
∀x ∈ E,
where PK (x) is informally defined as
PK (x) =

x
if x ∈ K ,
P (x)
if x ∈ K C .
Proof. Assume first that
||x − P (x)|| ≤ ||x − y|| ,
∀x ∈ K C , ∀y ∈ K ,
b If x ∈ K, then
Take x ∈ E.
hx − PK (x), z − PK (x)i = h0, z − xi = 0 ,
for any z ∈ K. Now if x ∈ K C , then for any y ∈ K, we have
||x − y||2 = ||x − P (x) − (y − P (x))||2
= hx − P (x) − (y − P (x)), x − P (x) − (y − P (x)i
= hx − P (x), x − P (x)i − 2hx − P (x), y − P (x)i + hy − P (x), y − P (x)i
= ||x − P (x)||2 + ||y − P (x)||2 − 2hx − P (x), y − P (x)i
←→ ||x − y||2 − ||x − P (x)||2 = ||y − P (x)||2 − 2hx − P (x), y − P (x)i ,
so that, by the assumption
2hx − P (x), y − P (x)i ≤ ||y − P (x)||2 .
Now take any z ∈ K and 0 < λ ≤ 1. Since K is convex, we have y = λz + (1 − λ)P (x) ∈ K.
Hence
2hx − P (x), λz + (1 − λ)P (x) − P (x)i ≤ ||λz + (1 − λ)P (x) − P (x)||2 ,
which implies, after deviding by λ,
2hx − P (x), z − P (x)i ≤ λ||z − P (x)||2 .
Letting λ → 0, we see that
2hx − PK (x), z − PK (x)i = 2hx − P (x), z − P (x)i ≤ 0 ,
so that the Bourbaki-Cheney-Goldstein inequality indeed holds in both cases.
Conversely, suppose
hx − PK (x), z − PK (x)i ≤ 0
b ∀z ∈ K .
∀x ∈ E,
If x ∈ K C , then we obtain for any y ∈ K
||y − x||2 = ||y − P (x) + P (x) − x||2 = hy − P (x) + P (x) − x, y − P (x) + P (x) − xi
= ||y − P (x)||2 + ||x − P (x)||2 − 2hy − P (x), x − P (x)i ≥ ||x − P (x)||2 ,
which completes the proof.
47
Lemma 4.29. The following is provable in RCA0 . Under the assumptions of Lemma 4.28
and the validity of the Bourbaki-Cheney-Goldstein inequality, we have
||z − PK (x)|| ≤ ||z − x|| ,
b ∀z ∈ K .
∀x ∈ E,
b and z ∈ K. Since
Proof. Suppose x ∈ E
hx − PK (x), z − PK (x)i ≤ 0 ,
we have
hx − PK (x), x − PK (x)i = hx − PK (x), z − PK (x) + x − zi ≤ hx − PK (x), x − zi ,
so that
0 ≤ ||x − PK (x)||2 ≤ hz − x, PK (x) − xi .
On the other hand, we also find
hz − PK (x), z − PK (x)i = hx − PK (x) + z − x, z − PK (x)i ≤ hz − x, z − PK (x)i ,
from which,
||z − PK (x)||2 ≤ hz − x, z − PK (x)i ≤ hz − x, z − PK (x)i + hz − x, PK (x) − xi = ||z − x||2 ,
i.e., the inequality we wanted to prove.
Lemma 4.30. The following is provable in RCA0 . Let K be a closed, convex and nonempty
b Suppose there exists a continuous P : dom(P ) → K, with
subset in a Hilbert space E.
C
b such that
K ⊆ dom(P ) ⊆ E,
P (x) = argz∈K min{||x − z|| : z ∈ K} ,
∀x ∈ K C ,
then the informally defined function PK attests to the definition of nonexpansiveness.
b By Lemma 4.28, we have
Proof. Let x, y ∈ E.
hx − PK (x), PK (y) − PK (x)i ≤ 0 ,
hy − PK (y), PK (x) − PK (y)i ≤ 0 .
Hence
hx − PK (x), PK (y) − PK (x)i + hPK (y) − y, PK (y) − PK (x)i
= hx − PK (x) + PK (y) − y, PK (y) − PK (x)i
= hPK (y) − PK (x), PK (y) − PK (x)i − hy − x, PK (y) − PK (x)i ≤ 0 .
Which implies, by the Cauchy-Schwarz inequality,
||PK (y) − PK (x)|| ≤ ||y − x|| ,
so that PK indeed attest to the definition of a nonexpansiveness.
48
When P attests to the assumption in Lemma 4.30, informally, PK denotes the metric
b to the closest point z ∈ K to x. We
b onto K, i.e., PK projects x ∈ E
projection of E
will show that a code for PK exists when K satisfies specific properties. For PK (x) to be
b it suffices for K to be convex:
uniquely defined for x ∈ E,
Lemma 4.31. The following is provable in RCA0 . Let K be a closed, convex and nonempty
b and suppose x ∈ E.
b If there exists y ∈ K such that
subset in a Hilbert space E
||x − y|| ≤ ||x − z|| ,
∀z ∈ K ,
then y ∈ K is uniquely determined.
Proof. Suppose there exists y 0 ∈ K such that ||x − y|| = ||x − y 0 ||. We need to show that
y = y 0 . Since K is convex, we have y/2 + y 0 /2 ∈ K, so that
x y 0 0 y
y
y
x
≤ − + − = ||x − y|| ,
+
||x − y|| ≤ x −
2
2
2 2 2
2 which implies
||y + y 0 − 2x|| = 2||x − y|| .
Hence, by the parallelogram law, we find
||y − y 0 ||2 = ||(y − x) − (y 0 − x)||2 = 2||y − x||2 + 2||y 0 − x||2 − ||y + y 0 − 2x||2
= 2||y − x||2 + 2||y − x||2 − 4||y − x||2 = 0 ,
i.e., y = y 0 .
One can easily find examples where K is a nonempty, closed and non-convex set in a
b such that there exists x ∈ E
b for which
Hilbert space E,
argz∈K min{||x − z|| : z ∈ K}
b with u 6= v and let
does not exist. For example, take any u, v ∈ E
K = {u} ∪ {v} ,
which denotes the complement of the open set
n
o n
o
b : x 6= u ∪ x ∈ E
b : x 6= v .
KC = x ∈ E
Clearly, K is a non-convex set. Moreover, we have
u + v
u + v
||u − v||
= =
−
u
−
v
,
2
2
2
so that
u + v
argz∈K min − z : z ∈ K
2
49
is not defined.
Up till now, it hasn’t been clear why we assumed K to be a closed set. This is because,
b the metric projection of E
b onto K
when K would be an open set in a Hilbert space E,
b For example, let us assume K 6= E
b is an
can never exist in the non-trivial case K 6= E.
C
b
open set in a Hilbert space E, take x ∈ K and suppose
z = argz∈K min{||x − z|| : z ∈ K} .
Since z ∈ K, there exists a ∈ E, r ∈ Q+ such that z ∈ B(a, r) ⊆ K. Now let λ > 0 be
such that
||a − z|| + λ < r ∧ λ < ||x − z|| .
Defining
y=z−
we find
z−x
λ,
||z − x||
z
−
x
λ ≤ ||a − z|| + λ < r ,
||a − y|| = a − z +
||z − x||
so that y ∈ B(a, r) ⊆ K. Moreover, we find
z − x (x − z)(||z − x|| − λ) ||x − y|| = x − z +
λ =
= ||z − x|| − λ < ||x − z|| ,
||z − x|| ||z − x||
a contradiction with the definition of z.
b → K of E
b
We are now ready to prove the existence of the metric projection PK : E
onto the closed ball K:
Theorem 4.32. (metric projection on the closed ball). The following is provable in RCA0 .
b Then there exists a nonexpansive
Let K = B(a, r) be a closed ball in a Hilbert space E.
b to the unique y ∈ K such that
function PK , which maps x ∈ E
||x − y|| = min{||x − z|| : z ∈ K} .
PK denotes the metric projection on K.
b after which we will define
Proof. We will start with informally defining PK (x) for x ∈ E,
b If x ∈ K, then cleary we must
a code for P as a continuous function. Hence, let x ∈ E.
have PK (x) = x. Now for x ∈
/ K, i.e., ||a − x|| > r, we claim that we must have
PK (x) = y := a − r
a−x
,
||a − x||
see Figure 11. Now
a
−
x
= r ,
||a − y|| = a − a − r
||a − x|| so that y ∈ K. We have
a − x (r − ||a − x||)(a − x) ||x − y|| = x − a − r
=
= ||a − x|| − r ,
||a − x|| ||a − x||
50
Moreover, for any z ∈ K, we have
||x − z|| ≥ ||a − x|| − ||a − z|| ≥ ||a − x|| − r = ||x − y|| .
Hence, by the Lemma 4.31, we conclude that


x
PK (x) =
a−x

a − r
||a − x||
if x ∈ K ,
if x ∈
/ K,
must be the required metric projection. Defining
P (x) = a − r
a−x
,
||a − x||
we know that there exists a code for P as a continuous function by the results of [1] Section
II.6. Moreover, we have
n
o
b : ||a − x|| =
K C ⊆ dom(P ) = x ∈ E
6 0 ,
so that by Lemma 4.30, Pk attests to the definition of nonexpansiveness.
It is left to show the existence of a code for PK as a continuous function within RCA0 .
Let ϕ1 denote the Σ01 formula
ϕ1 (u, s, v, t) ≡ (u, s), (v, t) ∈ E × Q+ ∧ ||u − a|| < r ∧ u = v ∧ s = t .
Where u = v can be seen as an equality in N, see also Section 2. Hence, we can write
ϕ1 (u, s, v, t) ≡ ∃n θ1 (n, u, s, v, t) ,
where θ1 is Σ00 . By Σ00 comprehension, we can define
Φ1 := {(n, u, s, v, t) ∈ N × E × Q+ × E × Q+ : θ1 (n, u, s, v, t)} .
Note that
(u, s)Φ1 (v, t) ↔ ∃n((n, u, s, v, t) ∈ Φ1 ) ↔ ∃n θ1 (n, u, s, v, t) ↔ ϕ1 (u, s, v, t) .
Now let ϕ2 denote the Σ01 formula
ϕ2 (u, s, v, t) ≡ (u, s), (v, t) ∈ E × Q+ ∧ ||u − a|| > r ∧ ||P (u) − v|| < s ∧ ds = t ,
where > 0 and d > 0 are any fixed real constants sufficing the following conditions:
• d > 1 + ,
• < 1.
51
Note that ϕ2 is indeed Σ01 , since
∃n ω1 (n) ∨∧ ∃m ω2 (m) ↔ ∃k((∃n ≤ k) ω1 (n) ∨∧ (∃m ≤ k) ω2 (m)) ,
where ω1 and ω2 are L2 -formulas. Moreover, any L2 -formula with of only existential number
quantifiers is Σ01 . Again, we can write
ϕ2 (u, s, v, t) ≡ ∃n θ2 (n, u, s, v, t) ,
where θ2 is Σ00 , and define
Φ2 := {(n, u, s, v, t) ∈ N × E × Q+ × E × Q+ : θ2 (n, u, s, v, t)} .
At least, we define the Σ01 formulas
ϕ3 (u0 , s0 , v 0 , t0 ) ≡ (∃(m, u, s, v, t) ∈ Φ1 ∪ Φ2 )(||u0 − u|| + s0 < s ∧ v = v 0 ∧ t = t0 )
≡ ∃n θ3 (n, u0 , s0 , v 0 , t0 ) ,
ϕ4 (u0 , s0 , v 0 , t0 ) ≡ (∃(m, u, s, v, t) ∈ Φ1 ∪ Φ2 ∪ Φ3 )(u = u0 ∧ s = s0 ∧ ||v 0 − v|| + t < t0 )
≡ ∃n θ4 (n, u0 , s0 , v 0 , t0 ) ,
where θ3 and θ4 are Σ00 , and
Φ3 := {(n, u0 , s0 , v 0 , t0 ) ∈ N × E × Q+ × E × Q+ : θ3 (n, u0 , s0 , v 0 , t0 )} ,
Φ4 := {(n, u0 , s0 , v 0 , t0 ) ∈ N × E × Q+ × E × Q+ : θ4 (n, u0 , s0 , v 0 , t0 )} .
Now, we define
Φ := Φ1 ∪ Φ2 ∪ Φ3 ∪ Φ4 ,
then Φ is a code for PK as a continuous function, see Theorem B.2 in the Appendix.
Figure 11: If x ∈
/ B(a, r) = K, then we get the metric projection PK (x) of x onto K by
subtracting the vector r(a − x)/||a − x|| with length r from a.
52
b is a complete separable metric
Definition 4.33. (narrowly coded boundary). Suppose E
b is defined by a weak boundary condition
space. If the closed set K ⊆ E
n
o
b
K := x ∈ E : f (x) ≤ r ,
b → R is a continuous function, then we define
where f : E
n
o
◦
b : f (x) < r ,
• K := x ∈ E
n
o
b : f (x) > r ,
• K C := x ∈ E
n
o
b : f (x) = r .
• ∂K := x ∈ E
It is easy to see that, by Theorem 2.14, all relevant sets exists within RCA0 . We say that
K has a narrowly coded boundary if
◦
(∀x ∈ ∂K)(∀ > 0)(∃u ∈ E) ||u − x|| < ∧ ∈ K ∪ K C .
If one investigates the proof of Theorem B.2 further, one will notice that we only needed
some specific properties of K and the existence of a code for a function P : dom(f ) → K,
b and ||x − P (x)|| ≤ ||x − y||, ∀x ∈ K C , ∀y ∈ K, to prove
such that K C ⊆ dom(P ) ⊆ E
b onto K. For example, the fact
the existence of a code Φ for the metric projection PK of E
that K has a narrowly coded boundary helps to show that the projection is also defined
on the boundary ∂K. Note that in general x ∈ ∂K can’t be defined by a Σ01 formula, even
if x ∈ E. Since, we did not explicitly made use of the definition of K and P at all, we
come to the following theorem:
Theorem 4.34. (metric projection in a Hilbert space). The following is provable in RCA0 .
b is a Hilbert space and let K be a nonempty, closed and convex subset in E
b
Suppose E
with a narrowly coded boundary. If there exists a continuous function P : dom(P ) → K,
such that
b,
• K C ⊆ dom(P ) ⊆ E
• ||x − P (x)|| ≤ ||x − y||, ∀x ∈ K C , ∀y ∈ K ,
Then there exists a code Φ for the continuous metric projection
b → K : x 7→ argy∈K min{||x − y|| : y ∈ K} ,
PK : E
and PK attests to the definition of nonexpansiveness.
The following theorem is inspired by [1] Lemma II.6.5.
b be a separable Banach
Theorem 4.35. The following is provable in RCA0 . Let E
b
b
space, K = B(a, r), a ∈ E, r > 0, a closed ball in E and hfn : n ∈ N, fn : K → Ei
a sequence of continuous functions. Suppose there exists a constant M > 0 such that
53
||fn (x)|| ≤P
M, ∀n ∈ N, x ∈ K. Then for any sequence hλn : n ∈ Ni of nonnegative reals
such that n λn converges, the function
X
f :=
λn fn
n
b
is
Pa well-defined continuous function f : K → E. If each fn , n ∈ N is nonexpansive and
n λn ≤ 1, then also f is nonexpansive.
Proof. Note that for any x ∈ K, the sequence
*
+
X
fn (x) : n ∈ N
k≤n
b by Theorem 2.10, since for m, n ∈ N, m < n, we have
indeed converges in E
n
X
X
X
λf
(x)
−
λ
f
(x)
=
λ
f
(x)
k
k k
k k
k≤n
k≤m
k=m+1
≤
n
X
λk ||fk (x)||
k=m+1
≤M
n
X
λk
k=m+1
≤M
∞
X
λk −
m
X
!
λk
m→∞
−−−→ 0 .
k=0
k=0
b be coded by Φn . Let ϕ denote
Let for n ∈ N, the continuous function λn fn : K → E
the Σ01 formula
ϕs (u, s, v, t) ≡ (u, s), (v, t) ∈ E × Q+ ∧ ||u − a|| < r
!!
∧ ∃m > 0 ∀k < m ∃vk ∈ E, tk ∈ Q+ (u, s)Φk (vk , tk )
∧v =
X
k<m
vk ∧
X
k<m
tk + M
∞
X
!
λk < t
k=m
≡ ∃n θs (n, u, s, v, t) ,
where θs is Σ00 . We define
Φs := (n, u, s, v, t) ∈ N × E × Q+ × E × Q+ : θs (n, u, s, v, t) ,
54
and the Σ10 formula
ϕe (u0 , s0 , v 0 , t0 ) ≡ ∃(m, u, s, v, t) ∈ Φs (||u − u0 || + s0 < s ∧ v 0 = v ∧ t0 = t)
≡ ∃n θe (n, u0 , s0 , v 0 , t0 ) ,
where θe is Σ00 . Defining
Φe := (n, u, s, v, t) ∈ N × E × Q+ × E × Q+ : θe (n, u, s, v, t) ,
we have that Φ := Φs ∪ Φe is a code for f as a continuous function, see Theorem B.5 in
the Appendix.
P
If each fn , n ∈ N is nonexpansive and n λn ≤ n, then for any x, y ∈ K, we have
X
X
||f (x) − f (y)|| ≤
λn ||fn (x) − fn (y)|| ≤
λn ||x − y|| ≤ ||x − y|| ,
n
n
so that f is indeed also nonexpansive.
Theorem 4.36. BGKH is equivalent to ACA0 over RCA0 .
Proof. There’s left to prove that BGKH implies ACA0 over RCA0 . We shall do this by
proving the monotone convergence theorem, i.e., the supremum of a given increasing
sequence of real numbers in [0, 1] exists. This statement is equivalent to ACA0 over RCA0
by [1] Theorem II.2.2.14
Hence, let hrn : n ∈ Ni be an increasing sequence in [0, 1]. We need to show that
supn rn exists. We will do this by defining a nonexpansive function f : K → K, where
b = l2 , such that ||a|| = supn rn for every fixed point a of f .
K is the closed unit ball in E
Moreover, f will have a unique fixed point.
Within RCA0 , we define a sequence of reals han : n ∈ Ni, where for n ∈ N
a0 = r 0 ;
q
2
an+1 = rn+1
− rn2 .
Inspired by the correspondence in Theorem 4.26, we define
αn = ha0 , . . . , an i ∈ l2 ,
and we have
||αn || =
n
X
!1/2
a20
= rn .
i=0
Hence, we need to prove that the sequence hαn : n ∈ Ni converges to some a ∈ l2 , then
||a|| = limn rn = supn rn as desired.
14
The proof that the general monotone convergence theorem, i.e., that any bounded increasing sequence
of real numbers is convergent, implies ACA0 over RCA0 , stays the same when we would restrict ourselves
to increasing sequences of real numbers in [0, 1].
55
Let l2 be coded as the completion of A. For any n ∈ N, we let ϕ(n, u, s, v, t) be the Σ01
formula
ϕs (n, u, s, v, t) ≡ (u, s), (v, t) ∈ A × Q+ ∧ ||u|| < 1 ∧ (lh(v) = max{lh(u), n + 1})
∧ t = 4s ∧ ∀m < n + 1 v(m) = pm(min{k∈N : 2−2k ≤ s2 /(n+1)})
∧ (∀m < lh(u)(m > n → v(m) = u(m)) ,
where, for m ∈ N,
am = hpmk : k ∈ Ni .
As before, we are able to define a set Φn,s ⊆ N × A × Q+ × A × Q+ , such that (u, s)Φn,s (v, t)
iff ϕs (n, u, s, v, t). We can then define the Σ01 formula
ϕe (n, u0 , s0 , v 0 , t0 ) ≡ ∃(m, u, s, v, t) ∈ Φn,s (||u − u0 || + s0 < s ∧ v 0 = v ∧ t0 = t) ,
and the set Φn,e ⊆ N × A × Q+ × A × Q+ , such that (u, s)Φn,e (v, t) iff ϕe (n, u, s, v, t). Next,
we define the Σ01 formula
ϕo (n, u0 , s0 , v 0 , t0 ) ≡ ∃(m, u, s, v, t) ∈ Φn,s ∪ Φe,s (u = u0 ∧ s = s ∧ ||v − v 0 || + t < t0 ) ,
and the set Φn,o ⊆ N × A × Q+ × A × Q+ , such that (u, s)Φn,o (v, t) iff ϕo (n, u, s, v, t). At
least, we define for each n ∈ N the set Φn := Φn,s ∪ Φn,e ∪ Φn,o . Then Φn codes a continuous
function Fn , with domain K, mapping x = xl : l ∈ N, xl = xl0 , . . . , xllh(xl )−1 ∈ A ∈ K
to the point liml Fn (xl ), where for n, l ∈ N

 a0 , . . . an , xl(n+1) , . . . , xl(lh(xl )−1) if lh(xl ) > n + 1
Fn (xl ) =
,
α
if
lh(x
)
≤
n
+
1
n
l
see Theorem B.6 in the Appendix. It should be clear that
||Fn (x)|| = lim ||Fn (xl )|| ≤ lim ||Fn (xl ) − αn || + ||αn || ≤ lim ||xl || + rn = ||x|| + rn ≤ 2 ,
l
l
l
and that for any y = yl : l ∈ N, yl = yl0 , . . . , yl(lh(yl )−1) ∈ A ∈ K,
||Fn (x) − Fn (y)|| = lim ||Fn (xl ) − Fn (yl )|| ≤ lim ||xl − yl || = ||x − y|| .
l
l
Hence, we conclude that hFn : n ∈ Ni denotes a sequence of uniformly bounded, nonexb so that by Theorem 4.35 the function
pansive functions Fn : K → E,
F :=
X 1
F
n+1 n
2
n
is well-defined and nonexpansive.
Now we define the continuous function f = PK ◦ F : K → K, where PK denotes the
metric projection PK on the closed ball K. As the composition of nonexpansive functions,
56
it is easy to see that f : K → K is also nonexpansive, and therefore, has a fixed point
x ∈ l2 by BKGH . We will show that x is a fixed point of F .
Note that for n, k ∈ N, we have

αk if n < k
Fk (αn ) =
,
α if n ≥ k
n
so that, since ||αn || = rn ≤ 1 for all n ∈ N,
∞
X
1
||F (αn ) − αn || = F
(α
)
−
α
k
n
n k+1
2
k=0
n
∞
X
X
1
1
= αn +
αk − αn k+1
k+1
2
2
k=0
k=n+1
∞
X
1
1
α
−
α
= 1 − n+1 αn +
n k+1 k
2
2
k=n+1
∞
X
||αk ||
1
||αn ||
≤ n+1 +
≤ n+1 +
k+1
2
2
2
k=n+1
n
X
1
1−
k+1
2
k=0
!
n→∞
−−−→ 0 .
Now suppose F (x) 6= x. Then we can choose n large enough such that
!
n
X
1
||x − F (x)||
1
<
,
||F (αn ) − αn || ≤ n+1 + 1 −
k+1
2
2
4
k=0
and we have
||F (x) − F (αn )||2 = hF (x) − F (αn ), F (x) − F (αn )i
= hF (x) − x + x − F (αn ), F (x) − x + x − F (αn )i
= ||F (αn ) − x||2 + ||x − F (x)||2 − 2hF (x) − x, F (αn ) − xi
≥ (||F (αn ) − αn || − ||αn − x||)2 + ||x − F (x)||2
− 2hF (x) − x, F (αn ) − xi
= ||F (αn ) − αn ||2 + ||αn − x||2 − 2||F (αn ) − αn || · ||αn − x||
+ ||x − F (x)||2 − 2hF (x) − x, F (αn ) − xi
≥ ||F (αn ) − αn ||2 + ||αn − x||2 − 4||F (αn ) − αn ||
+ ||x − F (x)||2 − 2hF (x) − x, F (αn ) − xi
> ||αn − x||2 − 2hF (x) − x, F (αn ) − xi
= ||αn − x||2 − 2hF (x) − PK (F (x)), F (αn ) − PK (F (x))i
≥ ||αn − x||2 ,
57
where the last inequality follows from the Bourbaki-Cheney-Goldstein inequality and the
fact that ||F (αn )|| ≤ 1, i.e., F (αn ) ∈ K, for all n ∈ N, which is easy to check. But then we
have ||F (x) − F (αn )|| > ||αn − x||, a contradiction, since F is nonexpansive. We conclude
that F (x) = x. Writing x = xn0 , . . . , xn(lh(xn )−1) : n ∈ N , we know from the proof of Theorem 4.26
that there exists a unique sequence hyn : n ∈ Ni = hhpnk : k ∈ Ni : n ∈ Ni of reals, with

= xkn
if lh(xk ) > n ,
pnk
= 0
if lh(xk ) ≤ n .
such that limn hy0 , . . . , yn i = x in l2 . Hence, we must have
lim F (hy0 , . . . , yn i) = F (x) = x = limhy0 , . . . , yn i .
n
n
Note that since ||hy0 , . . . , yn i|| is increasing in n, we have ||hy0 , . . . , xy i|| ≤ ||x|| ≤ 1, so
that F (hy0 , . . . , yn i) is indeed defined for each n ∈ N. Since for n ∈ N, we have
hy0 , . . . , yn i = hhp0k , . . . , pnk i : k ∈ Ni ,
we find for k ∈ N
Fk (hy0 , . . . , yn i) = lim Fk (hp0l , . . . , pnl i) =
l

ha0 , . . . , ak , yk+1 , . . . , yn i if n > k
α
if n ≤ k

ha0 − y0 , . . . , ak − yk i
if n ≥ k
k
,
and
Fk (hy0 , . . . , yn i) − hy0 , . . . , yn i =
.
ha − y , . . . , a − y , a , . . . , a i if n < k
0
0
n
n n+1
k
Hence, we have
0 = lim ||F (hy0 , . . . , yn i) − hy0 , . . . , yn i||
n
X 1
X 1
= lim Fk (hy0 , . . . , yn i) −
hy0 , . . . , yn i
k+1
k+1
n 2
2
k
k
X 1
= lim [F
(hy
,
.
.
.
,
y
i)
−
hy
,
.
.
.
,
y
i]
k
0
n
0
n k+1
n 2
k
X 1
X 1
= lim ha0 − y0 , . . . , ak − yk i +
ha0 − y0 , . . . , an − yn , an+1 , . . . , ak i
k+1
k+1
n 2
2
k>n
k≤n
s
= lim
n
X 1
X 1
2+
|a
−
y
|
|a |2 ,
k
k k
k k
2
2
k≤n
k>n
58
which is only possible if an = yn in R for all n ∈ N. Hence, we find that x is the limit
point of the sequence hαn : n ∈ Ni of points in l2 . This means that f : K → K has a
unique fixed point x, for which
||x|| = lim ||αn || = lim rn = sup rn ,
n
n
n
which shows that BGKH implies ACA0 over RCA0 .
Remark 4.37. Our method for proving BGKH →ACA0 consisted of the following steps
1. Define for each n ∈ N, the projection Fn : B(0, 1) → l2 , that takes the following
form in the ordinary mathematical l2
Fn : hx0 , x1 , . . . , xn , xn+1 , xn+2 , . . .i 7→ ha0 , a1 , . . . , an , xn+1 , xn+2 , . . .i .
2. Define the function F :=
P
1
k 2k+1 Fn
: B(0, 1) → l2 , which is nonexpansive.
3. By BGKH , let x be a fixed point of the nonexpansive function f := P B(0,1) ◦ F .
4. Proof that x is a fixed point of F .
5. Proof that ||x|| = limn ||ha0 , . . . , an i||.
It is interesting to compare these steps with strategies applied in other literature. Note
that the x we found in our proof, is a common fixed point of all Fn ’s, since for each n ∈ N
Fn (x) = lim Fn (ha0 , . . . ak i) = limha0 , . . . ak i = x .
k
k
Moreover, it is not to hard to see that if y is a common fixed point of all Fn ’s, then
y = limha0 , . . . , ak i ,
k
so that x is the only common fixed point of all Fn ’s. Hence, we actually needed to proof
that the Fn ’s have a common fixed point. Whereas in other literature, the knowledge of a
common fixed point of all Fn ’s in advance, is very useful for a lot of our performed steps.
F.e., let (in the ordinary mathematical sense) X be a strictly convex Banach space,
C a nonempty, closed and convex subset of X, hTn : n ∈ Ni a sequence of nonexpansive
functions Tn : CP
→ X with a common fixed point y and hλn : n ∈ Ni a sequence of positive
reals P
such that n λn = 1. Then in [9] Lemma 3, the common fixed point is used to show
that n λn Tn is well-defined in the ordinary mathematical sense, since for each n ∈ N and
x ∈ C, we have
||Tn (x)|| ≤ ||Tn (x) − Tn (y)|| + ||Tn (y)|| ≤ ||x − y|| + ||y|| ≤ diam(C) + ||y|| ,
so that
||T (x)|| ≤
X
λn ||Tn (x)|| ≤ diam(C) + ||y|| ,
n
P
i.e., n λn Tn (x) converges absolutely. We see that in the ordinary mathematical sense,
the knowledge that the Tn ’s are uniformly bounded for n ∈ N is enough to prove that T is
59
well-defined. This is also the way we’ve overcome the problem of having no knowledge of
a common fixed point of all Fn ’s in advance, since we did had knowledge of the definition
of these function themselves, and were able to prove that they are uniformly
By
Pbounded.
1
Fk was
Theorem 4.35, this also sufficed to prove in RCA0 that the function F = k 2k−1
well-defined within the context of reverse mathematics.
Also in [9] Lemma 3, the common fixed point y of all Tn ’s can be used to show that
that any fixed point of T is a common fixed point of all Tn ’s and vice versa. The fact
that a common fixed point of all Tn ’s is a fixed point of T is clear. Conversely, suppose
T (z) = z for z ∈ C. Note that for any w ∈ X, we have
X
X
||z − w|| = λn (Tk (z) − w) ≤
λn ||Tn (z) − w|| .
n
n
Now for w = y, we have ||Tn (z) − w|| = ||Tn (z) − Tn (w)|| ≤ ||z − w|| for all n ∈ N,
so that we must have ||Tn (z) − w|| = ||z − w|| for all n ∈ N, or we would find that
||z − w|| < ||z − w||, a contradiction. Hence, we both have
X
λn (Tn (z) − w) = ||z − w|| and ||Tn (z) − w|| = ||z − w|| , ∀n ∈ N .
n
Now, in a strictly convex Banach space, this would imply that Tn (z) − w = Tm (z) − w,
i.e., Tn (z) = Tm (z) for all n, m ∈ N, from which it is easy to see that z is a common fixed
point of all Tn ’s. Moreover, within the context of reverse mathematics, one can prove
that the same implication holds in RCA0 for a Hilbert space (which is always a strictly
b is a Hilbert space, hxn : n ∈ Ni a sequence of points in
convex Banach space). Since if E
b hλn : n ∈ Ni a sequence of positive reals such that P λn = 1, and we have
E,
n
X
λk xk = ||xn || , ∀n ∈ N ,
k
P
then, letting ||xn || = || k λk xk || =: a for all n ∈ N, we find
2
!
! *
+
X
X
X
X
X
a2 = λn xn ↔
λn a
λn a =
λn xn ,
λ n xn
n
n
↔
X
↔
X
n
n
n
2(λn λm a2 − hλn xn , λm xm i) = 0
n,m
(λn λm ||xn ||2 + λn λm ||xm ||2 − 2λn λm hxn , xm i) = 0
n,m
↔
X
hλn (xn − xm ), λm (xn − xm )i = 0
n,m
↔
X
λn λm hxn − xm , xn − xm i = 0
n,m
↔ hxn − xm , xn − xm i = 0 , ∀n, m ∈ N ,
60
which holds iff xn = xm for all n, m ∈ N. Hence, to generalize Bruck’s strategy for proving
that the fixed point x of F is indeed a common fixed point of all Fn ’s, it would suffice to
b such that ||Fn (x) − w|| ≤ ||x − w|| for all n ∈ N. However, we didn’t
find any point w ∈ E
find any such point,15 and directly proved that x = limn ha0 , . . . an i from the fact that x
was a fixed point of F , using our identification from Theorem 4.26.
The knowledge of a common fixed point of all Fn ’s in advance, could also be used to
show that the fixed point x of f is a fixed point of F , as in [4] Lemma 5.4 (moreover, the
knowledge of any fixed point of F in advance would suffice). The way to do this is very
similar as what we did in the proof of Theorem 4.36, by showing that if y is a common
fixed point of all Fn ’s and F (x) 6= x, then ||F (x) − F (y)|| > ||x − y||, which is impossible
if F is nonexpansive, so that F (x) = x. Intuitively, it should be clear why we were able
to generalize this inequality. Since the common fixed point y we are searching for should
actually be limn αn , it should be clear that the strict inequality ||F (x)−F (αn )|| > ||x−αn ||
should also hold for F (x) 6= x and n large enough.
4.3
b a uniformly convex Banach
The reverse mathematics for E
Space
We may conclude from Theorem 4.36 that that BGK implies ACA0 over RCA0 , since BGK
is stronger than BGKH . The main property of a Hilbert space that was used in proving
ACA0 → BGKH , was the parallelogram law applied in Lemma 4.10. Goebel’s article [10]
proves a similar lemma for ordinary uniformly convex Banach spaces, by using a modulus
of convexity. This will be the key for generalizing our proof to ACA0 → BGK.
b be a non-trivial uniformly convex Banach space with modulus of
Theorem 4.38. Let E
uniform convexity h. Then h(n) ≥ n for all n ∈ N.
b Then
Proof. Suppose n ∈ N and let 0 6= x ∈ E.
x/||x|| ∈ ∂B(0, 1)
and
(1 − 21−n )
x
∈ B(0, 1) .
||x||
Moreover, we have
x
x 1−n
1−n
,
||x|| − (1 − 2 ) ||x|| = 2
so that
−n
1−2
x/||x|| + (1 − 21−n x/||x||) ≤ 1 − 2−h(n) ,
= 2
and hence, n ≤ h(n).
b = {0}, we
Remark 4.39. Since BGK holds trivially for the separable Banach space E
b to be a non-trivial separable Banach space in the rest of this section.
may clearly assume E
Note than any function h : N → N is a modulus of uniform convexity for the trivial Banach
space {0}.
15
Of course, before we knew that x is indeed the limit point of the sequence hha0 , . . . , an i : n ∈ Ni.
61
b be a uniformly convex
Definition 4.40. The following definition is made in RCA0 . Let E
Banach space with modulus of uniform convexity h. For every q ∈ ]0, ∞[Q , we define

max n ∈ N : q ≤ 2−h(n)
if q ∈ 0, 2−h(0) Q ;
ĥ(q) :=
0
if q ∈ 2−h(0) , +∞ Q .
Note that by Theorem 4.38, the set {n ∈ N : q ≤ 2−h(n) } is finite for each q ∈ 0, 2−h(0) Q ,
since it is a subset of the set {n ∈ N : q ≤ 2−n }. Moreover, ĥ(q) is decreasing for
q ∈ ]0, +∞[Q .
b be a uniformly convex Banach space with modulus of uniform
Theorem 4.41. Let E
convexity h. Then ĥ diverges monotonically to infinity for q → 0+.
Proof. Let M ∈ N and q = min{2−h(0) , . . . , 2−h(M ) }. Then ĥ(q) ≥ M .
b be a uniformly convex
Lemma 4.42. The following is provable within RCA0 . Let E
b and r, R be constants
Banach space with modulus of uniform convexity h and let u, v ∈ E
b such that
such that 0 ≤ r < R. Suppose there exists an x ∈ E
u + v
||u − x|| < R , ||v − x|| < R , − x ≥ r .
2
Then for every q ∈ ]1 − r/R, +∞[Q , we have
||u − v|| < R · 21−ĥ(q) .
Proof. Suppose first q ∈ 2−h(0) , +∞ Q . Then we find
||u − v|| ≤ ||u − x|| + ||x − v|| < 2R = R · 21−ĥ(q) .
Now suppose 1 − r/R < 2−h(0) and q ∈ 1 − r/R, 2−h(0) Q . We have
u − x v − x (u − x) + (v − x) ≥ r .
R < 1 , R < 1 , R
2R
Now suppose ||u − v|| ≥ R · 21−ĥ(q) , i.e.,
u − v u − x v − x 1−ĥ(q)
.
R = R − R ≥ 2
By the uniform convexity, we then have
(u − x) + (v − x) ≤ 1 − 2−h(ĥ(q)) .
2R
But by the definition of ĥ, we find
1−
r
< q ≤ 2−h(ĥ(q)) ,
R
62
so that
(u − x) + (v − x) ≤ 1 − 2−h(ĥ(q)) < r ,
2R
R
a contradiction. Hence, we must have ||u − v|| < R · 21−ĥ(q) .
Lemma 4.43. The following is provable in RCA0 . Suppose K is a (separably) closed
b f : K → K is nonexpansive and that
subset of the uniformly convex Banach space E,
there exists a positive constant L such that for all x, y ∈ K, ||x − y|| ≤ L. Suppose that
x, y, a := (x + y)/2 ∈√K, and that < L is such that ||x − F (x)|| < , ||y − F (y)|| < .
Then for every q ∈ ]2 , +∞[Q , we have
√
+ ,
2
||a − F (a)|| < max
Proof. Suppose first ||x − y|| <
√
L
1−ĥ(q)
.
+ 2
2
. Then
√
y − x +<
+ .
||a − F (a)|| = ||a − x|| + ||x − F (x)|| < 2
2
√
Now suppose ||x − y|| ≥ . Defining
1
1
r := ||x − y||, R := ||x − y|| + and u = a, v = F (a) ,
2
2
we have
1
||x − y|| + √
2
r
≤ 2 .
=
1− =1− 2
1
R
||x − y||
||x − y||
2
In an analogous way as in Lemma 4.11, we have
1−ĥ(q)
||a − F (a)|| < R · 2
≤
L
+ 21−ĥ(q) ,
2
for every
√ q ∈ ]1 − r/R, +∞[Q by Lemma 4.42. Then this inequality holds also for every
q ∈ ]2 , +∞[Q .
b be a separale Banach space, K
Lemma 4.44. The following is provable in ACA0 . Let E
b and g : K → R continuous. Then the set
a separably closed in E
Z = (a, q, r) ∈ E × Q+ × Q+ : ∃x ∈ BEb (a, q) (g(x) < r)
exists.
63
Proof. Let Φ ⊆ N × E × Q+ × Q × Q+ be the code for g. Then the property
ϕ ≡ ∃x ∈ BEb (a, q) (g(x) < r)
can be expressed by the following arithmetical formula:
ψ ≡ ∃(n, a0 , q 0 , b, r0 ) ∈ N × E × Q+ × Q × Q+

ψ1
z
}|
{

0 0
0
0 0
((n, a , q , b, r ) ∈ Φ) ∧ BEb (a , q ) ∩ K 6= ∅


∧ BEb (a0 , q 0 ) ⊆ BEb (a, q) ∧ BR (b, r0 ) ⊆ ] − ∞, r[ 
,
{z
} |
|
{z
}
ψ2
ψ3
Note that this formula is indeed arithmetical, moreover, we have
ψ1 ↔ ∃n||xn − a0 || < q 0 ;
ψ2 ↔ ||a − a0 || + q 0 ≤ q ;
ψ3 ↔ b + r0 < r ,
where hxn : n ∈ Ni is a code for K.
b induces
To be able to work in the relative topology that a separable Banach space E
b in our proof, we need a concept of open and closed
on a separably closed set K in E,
sets in K. These are very similar defined as ordinary open and closed sets in a separable
metric space.
b be a complete separable metric space. Let K be a
Definition 4.45. Within RCA0 , let E
b defined by the sequence hxn : n ∈ Ni of points in E.
b A (code
separably closed set in E,
+
for an) open set U in K is a set U ⊆ N × N × Q . A point x ∈ K is said to belong to K,
denoted by x ∈ K, if
∃n ∃m ∃r (d(x, xm ) < r ∧ (n, m, r) ∈ U ) .
A closed set in K is the complement of an open set in K. I.e., a code for a closed set C is
the the same thing as a code for an open set U , but we define x ∈ C iff x ∈
/ U.
By this definition, we are able to easily make the necessary adjustments to the following
two theorems:
Theorem 4.46. (Banach’s contraction principle). The following is provable in RCA0 .
b and
Suppose K is a nonempty and separably closed subset of the separable Banach space E
f : K → K is a contraction, i.e., a Lipschitz-continuous function with Lipschitz-constant
0 ≤ L < 1. Then f has a unique fixed point.
64
Proof. The proof goes exactly the same as in Subsection 4.2.1. We only need to make use
b then
of the fact that if hyn : n ∈ Ni is a sequence in K that converges to some y ∈ E,
y ∈ K. This follows easily from the definition of a separably closed set.
Theorem 4.47. (Cantor’s intersection theorem). The following is provable in RCA0 . Let
b be a complete separable metric space. Let hCn : n ∈ Ni be a decreasing sequence of
E
b Suppose there exists a decreasing
nonempty closed sets in a separably closed set K in E.
sequence of reals hn : n ∈ Ni with limn n = 0 such that u, v ∈ Cn implies ||u − v|| ≤ n .
If there exists a sequence of points hxn : n ∈ N, xn ∈ Cn i, then this sequence converges to
the unique point x ∈ K such that x ∈ Cn for all n ∈ N.
Proof. Again, one only needs the fact that the limit point of hxn : n ∈ Ni is contained in
K to generalize the proof from Subsection 4.2.1.
At least, one needs a statement similar to Theorem 2.14, but restricted to open sets in
a separably closed set.
Lemma 4.48. The following is provable in RCA0 . Let for any n ∈ N, ϕ(x, n) be a Σ01
b be a complete separable metric space and K a separably closed set
formula, and let E
b Assume that for all n ∈ N and x, y ∈ K, (x = y) ∧ ϕ(x, n) implies ϕ(y, n). Then
in E.
there exists a sequence of (codes for) open sets hUn : n ∈ Ni in K, such that for all
x ∈ K, x ∈ Un if and only if ϕ(x, n).
Theorem 4.49. ACA0 is equivalent to BGK over RCA0 .
Proof. By Theorem 4.36, BGK for the Hilbert space l2 implies ACA0 . Hence, we only
need to consider the proof of ACA0 → BGK.
b be a uniformly convex Banach space with modulus of uniform convexity h, K a
Let E
b and f : K → K nonexpansive.
nonempty, bounded, separably closed and convex set in E
Suppose ||x|| < M for all x ∈ K. W.l.o.g., we may assume 1 ≤ M ∈ N.
Analogous to the proof of Theorem 4.20, we are able to find a decreasing sequence of
codes for nonempty open sets hQn : n ∈ N, n ≥ 2i, where Qn denotes the informal open set
2M
Qn = x ∈ K : ||x − f (x)|| <
,
n
with n ∈ N and n ≥ 2. As in the proof of 4.20, we can within ACA0 define for every
n ∈ N, n ≥ 2,
dn = inf{||x|| : x ∈ Qn } ,
by Lemma 4.44, and we have
d2 ≤ d3 ≤ . . . ≤ dn ≤ . . . , with di ≤ M for all i ≥ 2 .
Hence, ACA0 implies the existence of
d = sup{dn : n ∈ N} ,
and we have limn dn = d ≤ M .
We define the sequence of natural numbers
λ : N → N : n 7→ (M + 1)22 max{h(0),...,h(n)}+3 ,
then λ is an increasing sequence, satisfying the following three properties:
65
1. λ(n) > 8n2 for all n ∈ N, n ≥ 2.
Take n ∈ N, n ≥ 2. Since h(n) ≥ n by Theorem 4.38, we find
λ(n) > 22h(n)+3 ≥ 8 · 22n > 8n2 .
2. 22−n /λ(n) < 1/n for all n ∈ N, n ≥ 2.
Take n ∈ N, n ≥ 2. Then as before, we find
λ(n) > 8 · 22n > 4 · 22n = 2n · 2n+2 > n22−n .
3. λ(n) > M 22h(k)+3 for all n, k ∈ N, n ≥ 2, k ≤ n.
This is clear from the definition of λ.
Again, for every n ∈ N, n ≥ 2, there exists a code for the open set
1
An = Qλ(n) ∩ B 0, d +
⊆K,
n
and hAn : n ∈ N, n ≥ 2i defines a decreasing sequence of nested, nonempty, open sets.
Now for u, v ∈ An , we have
||u − 0|| = ||u|| < d +
1
1
and ||v − 0|| = ||v|| < d + .
n
n
(3)
Also, since u, v ∈ Qλ(n) ,
||u − f (u)|| <
2M
2M
and ||v − f (v)|| <
.
λ(n)
λ(n)
Now Lemma 4.43, where u, v are points in the convex set K and hence (u + v)/2 ∈ K,
implies
(s
)
u + v
u
+
v
M
2M
2M
< max
+
, M+
21−ĥ(q) ,
2 − f
2
2λ(n) λ(n)
λ(n)
ip
h
8M/λ(n), +∞ . Now for every n ∈ N, n ≥ 2, we have
for all q ∈
Q
s
M
2M
M
2M
M
M
2M
+ 2 <
+
<√
+
<
.
2
2λ(n) λ(n)
8n
4n 4n
n
2 · 8n
Also, since
λ(n) > M 22h(k)+3 ,
i.e.,
s
2−h(k) >
66
8M
,
λ(n)
for all k ∈ N, k ≤ n, we can take
#s
q = min 2−h(0) , . . . , 2−h(n) ∈
8M
, +∞
λ(n)
"
,
Q
with ĥ(q) ≥ n as in Theorem 4.41, and since 21−n ≤ 1/n for all n ∈ N, n ≥ 2, we find
2M
2M
M
22−n M
M
M
2M
1−ĥ(q)
M+
2
≤ M+
21−n ≤
+
<
+
=
.
λ(n)
λ(n)
n
λ(n)
n
n
n
Hence, we conclude (u + v)/2 ∈ Qn , so that
u + v
u + v 2 − 0 = 2 ≥ dn ,
and Lemma 4.42, together with (3) and (4), implies
dn
1
1−ĥ(q)
, +∞ .
·2
, ∀q ∈ 1 −
||u − v|| < d +
n
d + 1/n
Now for every i, k ∈ N, we define within ACA0
d(i) = min q ∈ Q : d + 2−(i+1) < q < d + 2−i ,
N
(i)
dk = min q ∈ Q : dk − 2−i < q < dk − 2−(i+1) .
N
It is easy to see that
d =R d(i) : i ∈ N, d(i) ∈ Q ,
D
E
(i)
(i)
dk =R dk : i ∈ N, dk ∈ Q ,
for every k ∈ N, and that for all i, j ∈ N, with i > j,
(j)
(i)
d
dk
dk
i→∞
< (i) k
−−−→
.
(j)
d + 1/n
d + 1/n
d + 1/n
Moreover
(n)
dn
dn
<
,
(n)
d + 1/n
d + 1/n
for all n ∈ N, with
(n)
dn
dn
lim
= lim
= 1.
n→∞ d(n) + 1/n
n→∞ d + 1/n
Hence, for all n ∈ N, n ≥ 2, and for all u, v ∈ An , we find
(n)
1
1−ĥ 1−dn /(d(n) +1/n)
||u − v|| < n := d +
·2
,
n
67
(4)
with
(n)
dn
lim ĥ 1 − (n)
n→∞
d + 1/n
!
= +∞
by Theorem 4.41, so that limn n = 0.
Note that again, one can easily extract a sequence of points hxn : n ∈ N, xn ∈ An i. As
in the proof of Theorem 4.20, this implies the existence of a point y contained in all open
sets An for n ≥ 2, which is a fixed point of f . We may hence conclude that ACA0 implies
BGK over ACA0 .
68
A
Appendix: Nederlandstalige samenvatting
In [1] analyseert Simpson Brouwers fixpuntstelling in zijn hoofdstuk over het zwakke lemma
van König, waarin hij bewijst dat Brouwers fixpuntstelling equivalent is aan het zwakke
lemma van König over RCA0 . In dit werk bekijken we een andere fixpuntstelling, namelijk
de fixpuntstelling van Browder-Göhde-Kirk. Het doel van deze thesis is het uitvoeren en
analyseren van de reverse mathematics van deze stelling.
Om dit werk zo goed als mogelijk zelfomvattend te maken, vangen we aan met het
definiëren van de taal L2 van de tweede orde rekenkunde en het formeel systeem Z2
in Hoofdstuk 1. Vervolgens bespreken we de natuurlijke deelsystemen RCA0 , WKL0 ,
ACA0 , ATR0 en Π11 -CA0 van Z2 in Hoofdstuk 2. RCA0 is het zwakste systeem van deze
deelsystemen, i.e., is bevat in alle andere deelsystemen. We bespreken de basisconcepten
van reële analyse en topologie die in dit deelsysteem, en dus eveneens in alle andere
deelsystemen, ontwikkeld kunnen worden in Hoofstuk 2. Hoofdstukken 1 en 2 zijn
voornamelijk gebaseerd op, of bevatten notities van [1] en [2].
Het doel van Hoofdstuk 3 is het introduceren van het concept reverse mathematics. De
Hoofdvraag luidt: ‘Gegeven een standaardstelling ϕ uit de wiskunde, wat is het zwakste
deelsysteem van Z2 waarin we ϕ kunnen bewijzen? ’ Als voorbeeld tonen we aan dat
Brouwers fixpuntstelling voor het eenheidsvierkant equivalent is aan WKL0 over RCA0 . Dit
bewijs is hoofdzakelijk gebaseerd op Simpsons bewijs van de meer algemene fixpuntstelling
van Brouwer in [1] Sectie IV.7.
In Hoofdstuk 4 bekijken we de andere fixpuntstelling, deze van Browder-Göhde-Kirk.
We tonen aan dat deze stelling, ook wel genoteerd BGK, equivalent is aan ACA0 over
RCA0 . De richting ACA0 → BGK is hoofdzakelijk gebaseerd op een veralgemening van een
standaardbewijs uit [3], d.m.v. onderzoek naar hoe en op welke manier we deze strategieën
kunnen vertalen naar de context van reverse mathematics. Het bewijs van de omgekeerde
richting BGK→ ACA0 is voornamelijk gebaseerd op ideeën uit [4].
69
B
Appendix: Coding continuous functions
B.1
Coding the metric projection on a closed ball in a Hilbert
space.
b be a separable Banach space.
Lemma B.1. The following is provable in RCA0 . Let E
b
Consider the following subsets of E:
• K = B(a, r) ,
◦
• K = B(a, r) ,
◦
• ∂K = (K ∪ K C )C .
◦
Then for every x ∈ ∂K and > 0, there exists u ∈ (K ∪ K C ) ∩ E with ||u − x|| < .
Proof. Let x ∈ ∂K, i.e., ||x − a|| = r, and let > 0. W.l.o.g., we may assume that r < 2.
Let u ∈ E be such that
r
(x
−
a)
< .
u − x −
2
2
Then
||u − x|| <
r
+ ||x − a|| = ,
2
2
and
r
(x
−
a)
r
(x
−
a)
+ x −
||u − a|| ≤ u − x −
− a
2
2
(2 − r)(x − a) = +r− = r,
< + 2
2
2
2
◦
i.e., u ∈ K ∩ E.
Theorem B.2. The following is provable in RCA0 . Let Φ be as in Theorem 4.32, then Φ
defines a code for PK as a continuous function.
Proof. First we need to check that Φ indeed defines a continuous functions, i.e., that the
properties of Definition 2.15 are fulfilled. Note that
(u, s)Φ(v, t) ↔ (u, s)Φ2 (v, t) ∨ (u, s)Φ1 (v, t) ∨ (u, s)Φ3 (v, t) ∨ (u, s)Φ4 (v, t) .
1. Suppose (u, s)Φ(v, t)∧(u, s)Φ(v 0 , t0 ). Note that we do not need to check every possible
case by symmetry and the fact that some cases, such as (u, s)Φ1 (v, t) ∧ (u, s)Φ2 (v 0 , t0 ),
are impossible.
• Suppose (u, s)Φ1 (v, t) ∧ (u, s)Φ1 (v 0 , t0 ), then
||v − v 0 || = 0 < t + t0 .
70
• Suppose (u, s)Φ2 (v, t) ∧ (u, s)Φ2 (v 0 , t0 ), then
||v − v 0 || ≤ ||P (u) − v|| + ||P (u) − v 0 || < 2s < 2ds = t + t0 .
• Suppose (u, s)Φ1 (v, t) ∧ (u, s)Φ3 (v 0 , t0 ). Then we have either (u0 , s0 )Φ1 (v 0 , t0 ) or
(u0 , s0 )Φ2 (v 0 , t0 ) such that ||u − u0 || + s < s0 . In the first case, we have
||v − v 0 || = ||u − u0 || < s0 − s = t0 − t < t + t0 .
In the second case, we find
||v − v 0 || ≤ ||v − u|| + ||u − P (u0 )|| + ||P (u0 ) − v 0 ||
< 0 + ||u − u0 || + s0
(Lemma 4.29)
< s0 − s + s0
=
1+ 0
t − t < t + t0 ,
d
since 1 + < d.
• Suppose (u, s)Φ2 (v, t) ∧ (u, s)Φ3 (v 0 , t0 ). Then we have either (u0 , s0 )Φ1 (v 0 , t0 ) or
(u0 , s0 )Φ2 (v 0 , t0 ) such that ||u − u0 || + s < s0 . In the first case, we have
||v − v 0 || = ||v − u0 ||
≤ ||v − P (u)|| + ||P (u) − u0 ||
< s + ||u − u0 ||
(Lemma 4.29)
< s + s0 − s
= ( − 1)s + t0 < t + t0 ,
since < 1. In the second case, we find
||v − v 0 || ≤ ||v − P (u)|| + ||P (u) − P (u0 )|| + ||P (u0 ) − v 0 ||
< s + ||u − u0 || + s0
(Lemma 4.30)
< s + s0 − s + s0
< (1 + )s0 + ( − 1)s
=
1+ 0 −1
t +
t < t + t0 ,
d
d
since 1 + < d.
• Suppose (u, s)Φ3 (v, t) ∧ (u, s)Φ3 (v 0 , t0 ). Then we have either (u0 , s0 )Φ1 (v, t)
or (u0 , s0 )Φ2 (v, t) such that ||u − u0 || + s < s0 and either (ũ, s̃)Φ1 (v 0 , t0 ) or
(ũ, s̃)Φ2 (v 0 , t0 ) such that ||u − ũ|| + s < s̃.
71
− Suppose (u0 , s0 )Φ1 (v, t) ∧ (ũ, s̃)Φ1 (v 0 , t0 ). Then we find
||v − v 0 || = ||u0 − ũ|| ≤ ||u0 − u|| + ||u − ũ|| < s0 − s + s̃ − s < t + t0 .
− Suppose (u0 , s0 )Φ1 (v, t) ∧ (ũ, s̃)Φ2 (v 0 , t0 ). Then we find
||v − v 0 || ≤ ||v − u0 || + ||u0 − P (ũ)|| + ||P (ũ) − v 0 ||
< 0 + ||u0 − ũ|| + s̃
(Lemma 4.29)
≤ ||u0 − u|| + ||u − ũ|| + s̃
< s0 − s + s̃ − s + s̃
=t+
1+ 0
t − 2s < t + t0 ,
d
since d > 1 + .
− Suppose (u0 , s0 )Φ2 (v, t) ∧ (ũ, s̃)Φ2 (v 0 , t0 ). Then we find
||v − v 0 || ≤ ||v − P (u0 )|| + ||P (u0 ) − P (ũ)|| + ||P (ũ) − v 0 ||
< s0 + ||u0 − ũ|| + s̃
(Lemma 4.30)
≤ s0 + ||u0 − u|| + ||u − ũ|| + s̃
< s0 + s0 − s + s̃ − s + s̃
=
+1
1+ 0
t+
t − 2s < t + t0 ,
d
d
since d > 1 + .
• Suppose that we have (u, s)Φ1 (v, t)∧(u, s)Φ4 (v 0 , t0 ), (u, s)Φ2 (v, t)∧(u, s)Φ4 (v 0 , t0 )
0 0
or (u, s)Φ3 (v,
must have either (u, s)Φ1 ṽ, t̃ ,
t) ∧ (u, s)Φ4 (v, t ). Then we
(u, s)Φ2 ṽ, t̃ or (u, s)Φ3 ṽ, t̃ such that ||v 0 − ṽ|| + t̃ < t0 . In any case, we know
already that we have
||v − ṽ|| ≤ t + t̃ ,
so that
||v − v 0 || ≤ ||v − ṽ|| + ||ṽ − v 0 || < t + t̃ + t0 − t̃ = t + t0 .
0 0
• At least, suppose
(u, s)Φ4 (v,
t) ∧ (u, s)Φ4 (v , t ). Then we must have either
(u, s)Φ1 ṽ, t̃ , (u, s)Φ2 ṽ, t̃ or (u, s)Φ3 ṽ, t̃ with ||v − ṽ|| + t̃ < t, and either
(u, s)Φ1 v̂, t̂ , (u, s)Φ2 v̂, t̂ or (u, s)Φ3 v̂, t̂ such that ||v 0 − v̂|| + t̂ < t0 . In
any case, we know already that we have
||ṽ − v̂|| ≤ t̃ + t̂ ,
so that
||v − v 0 || ≤ ||v − ṽ|| + ||ṽ − v̂|| + ||v̂ − v 0 || < t − t̃ + t̃ + t̂ + t0 − t̂ = t + t0 .
72
2. Suppose (u, s)Φ(v, t) ∧ ||u − u0 || + s0 < s, with u0 ∈ E, s0 ∈ Q+ .
• Let (u, s)Φ1 (v, t) ∨ (u, s)Φ2 (v, t). Then one easily sees that ϕ3 (u0 , s0 , v, t), so
that (u0 , s0 )Φ3 (v, t) and hence (u, s)Φ(v, t)
• Let (u, s)Φ3 (v, t), i.e, there exists (m, ũ, s̃, v, t) ∈ Φ1 ∪ Φ2 with ||ũ − u|| + s < s̃.
Then we have
||u0 − ũ|| + s0 ≤ ||u0 − u|| + ||u − ũ|| + s0 < ||u − ũ|| + s < s̃ ,
so that ϕ3 (u0 , s0 , v, t) and (u0 , s0 )Φ(v, t).
• Let (u, s)Φ4 (v, t), i.e, there exists m, u, s, ṽ, t̃ ∈ Φ1 ∪Φ2 ∪Φ3 with ||ṽ−v||+t̃ < t.
− If (u, s)Φ1 (ṽ, t̃) ∨ (u, s)Φ2 (ṽ, t̃), then (u0 , s0 )Φ3 ṽ, t̃ which implies we have
(u0 , s0 )Φ4 (v, t), so that (u0 , s0 )Φ(v, t).
− If now (u, s)Φ3 ṽ, t̃ , then there exists k, ũ, s̃, ṽ, t̃ ∈ Φ1 ∪ Φ2 such that
||ũ − u|| + s < s̃, and we have
||u0 − ũ|| + s0 ≤ ||u0 − u|| + ||u − ũ|| + s0 < ||u − ũ|| + s < s̃ ,
so that we again find (u0 , s0 )Φ3 (ṽ, t̃) and hence (u0 , s0 )Φ4 (v, t), so that
(u0 , s0 )Φ(v, t).
3. Suppose (u, s)Φ(v, t) ∧ ||v − v 0 || + t < t0 , with v 0 ∈ E, t0 ∈ Q+ .
• Let (u, s)Φ1 (v, t) ∨ (u, s)Φ2 (v, t) ∨ (u, s)Φ3 (v, t). Then one easily sees that
ϕ4 (u, s, v 0 , t0 ), so that (u, s)Φ4 (v 0 , t0 ) and hence (u, s)Φ(v 0 , t0 )
• Let (u, s)Φ4 (v, t), i.e, there exists (m, u, s, ṽ, t̃) ∈ Φ1 ∪Φ2 ∪Φ3 with ||ṽ−v||+ t̃ < t.
Then we have
||v 0 − ṽ|| + t̃ ≤ ||v − v 0 || + ||v − ṽ|| + t̃ < ||v − v 0 || + t < t0 ,
so that ϕ4 (u, s, v 0 , t0 ) and (u, s)Φ(v 0 , t0 ).
Hence, Φ defines a continuous function f in the context of reverse mathematics.
b We will do this in two steps.
We need to show that f (x) = Pk (x) for all x ∈ E.
b
1. dom(f ) = E
b and > 0. Let s ∈ Q+ be such that s < /d and u ∈ E such
Take any x ∈ E
◦
that ||u − x|| < s. If u ∈ K, then we have (u, s)Φ1 (v, t) with t = s < /d < . If
◦
u∈
/ K, then we have (u, s)Φ2(v, t) with t = ds < . Note that, if x ∈ K ∨ x ∈
/ K, we
◦
◦
can always find such a u ∈ K ∪ K C ∩ E, since the sets K and K C are open. If
◦
C
x ∈ ∂K, we can find such a u ∈ K ∪ K ∩ E by Lemma B.1.
2. f (x) = PK (x) for all x ∈ dom(f )
b we need to show that f (x) = PK (x). Hence, let
Suppose x ∈ dom(f ) = E,
(u, s)Φ(v, t) and ||x − u|| < s, we need to show that ||PK (x) − v|| ≤ t.
73
• If (u, s)Φ1 (v, t), then we have u = v, s = t and
||PK (x) − v|| ≤ ||x − v|| = ||x − u|| < s = t .
• If (u, s)Φ2 (v, t), then ||P (u) − v|| < s, t = ds and we find
||PK (x)−v|| ≤ ||PK (x)−P (u)||+||P (u)−v|| < ||x−u||+s < (1+)s < ds = t .
• If (u, s)Φ3 (v, t), then we have either (u0 , s0 )Φ1 (v, t) or (u0 , s0 )Φ2 (v, t) such that
||u − u0 || + s < s0 . Since
||x − u0 || ≤ ||x − u|| + ||u − x|| < s + s0 − s = s0
we know already that we have ||PK (x) − v|| ≤ t in either case.
• If (u, s)Φ4 (v, t), we have either (u, s)Φ1 (v 0 , t0 ), (u, s)Φ2 (v 0 , t0 ) or (u, s)Φ3 (v 0 , t0 )
such that ||v − v 0 || + t0 < t. In any case, we know that ||PK (x) − v 0 || ≤ t0 , so
that
||PK (x) − v|| ≤ ||PK (x) − v 0 || + ||v 0 − v|| < t + t − t0 = t .
Hence, we conclude that Φ is a code for the continuous function PK .
B.2
Coding the metric projection on a half-space in a Hilbert
space
The generality of our proof of the existence of the metric projection on the closed ball
in a Hilbert space, makes us able to prove, within RCA0 , the existence of many more
metric projections on nonempty, closed and convex sets in Hilbert spaces. This is stated
in Theorem 4.34. For example, the same strategy for defining the code can be applied
to show that the metric projection on a half-space exists within RCA0 as a continuous
function. These projections are also considered in Neumann’s article [4].
b be a Hilbert space and by Theorem 2.14, let for
Definition B.3. (half-space). Let E
b a 6= 0, r ∈ R
a ∈ E,
b : ha, xi ≤ r} .
H(a,r) := {x ∈ E
b We call H(a,r) a half-space in E.
b It is easy to see that any
Then H(a,r) is a closed set in E.
b i.e., E
b 6= {0}, is convex and nonempty.
half-space in a nontrivial Hilbert space E,
Theorem B.4. (metric projection on a half-space). The following is provable in RCA0 . Let
b be a Hilbert space and K = H(a,r) a half-space in E.
b Then there exists a nonexpansive
E
b to the unique y ∈ K such that
function PK , which maps x ∈ E
||x − y|| = min{||x − z|| : z ∈ K} .
PK denotes the metric projection on K.
74
b is a nontrivial Hilbert space, so that K is a nonempty,
Proof. We may clearly assume that E
b
closed and convex set in E. We will apply Theorem 4.34 to prove the existence of a code
for PK . Hence, let x ∈ K C , i.e., ha, xi > r, we claim that we must have
PK (x) = y := x −
We have
ha, yi = ha, xi −
ha, xi − r
a.
||a||2
ha, xi − r
ha, ai = r ,
||a||2
so that y ∈ K. Defining informally


x
PK (x) =
ha, xi − r

P (x) := x −
a
||a||2
if x ∈ K ,
if x ∈
/ K,
b → E,
b we have for any
where we know there exists a code for the function P : E
x ∈ K C , z ∈ K,
ha, xi − r
ha, xi − r
hx − PK (x), z − PK (x)i =
a, z − P (x) =
(ha, zi − ha, P (x)i)
2
||a||
||a||2
=
ha, xi − r
(ha, zi − r) ≤ 0 ,
||a||2
which denotes the Bourbaki-Cheney-Goldstein inequality (the inequality is trivial for
x ∈ K), so that by Lemma 4.28, we have
||x − P (x)|| ≤ ||x − z|| ,
∀x ∈ K C , ∀z ∈ K .
Hence, by Lemma 4.31, we conclude that PK is the required metric projection.
It is left to show the existence of a code for PK as a continuous function within RCA0 .
Defining
◦
b : ha, xi < r} ,
• K := {x ∈ E
b : ha, xi > r} ,
• K C := {x ∈ E
b : ha, xi = r} ,
• ∂K := {x ∈ E
this will follow
◦ from Theorem 4.34 if we can show that for every x ∈ ∂K and > 0, there
exists u ∈ K ∪ K C ∩ E such that ||x − u|| < , i.e., K has a narrowly coded boundary.
b be such that ha, xi = r and let > 0. Since a 6= 0, we can find λ > 0 such
Hence let x ∈ E
that
||a||
λ<
.
2
Now let u ∈ E be such that
u − x + λa < min , λ
,
||a||2 2 ||a||
75
then
||u − x|| <
λ
+
< + = ,
2 ||a||
2 2
and
λa
λa
ha, ui = ha, xi + a, u − x +
+ a,
||a||2
||a||2
λa
.
= r + λ + a, u − x +
||a||2
Now by the Cauchy Schwars inequality, we have
a, u − x + λa
≤ ||a|| u − x + λa < ||a|| λ = λ ,
||a||2
||a||2 ||a||
in particular, we have
λa
6= −λ ,
a, u − x +
||a||2
so that ha, ui =
6 r
B.3
Coding of a function series in a separable Banach space
Theorem B.5. The following is provable in RCA0 . Let Φ be as in Theorem 4.35, then Φ
defines a code for f as a continuous function.
Proof. We will first check that Φ indeed codes a continuous function by checking the
properties of Definition 2.15. Note that we have (u, s)Φ(v, t) ↔ ϕs (u, s, v, t) ∨ ϕe (u, s, v, t).
1. Suppose (u, s)Φ(v, t) and (u, s)Φ(v 0 , t0 ).
• Suppose (u, s)Φs (v, t) and (u, s)Φs (v 0 , t0 ). Then there exists natural numbers
m, m0 ∈ N+ , (vk , tk ) ∈ E × Q+ , ∀k < m, (vl0 , t0l ) ∈ E × Q+ , ∀l < m0 , such that
v=
X
k<m
vk ∧
X
tk + M
k<m
∞
X
0
λk < t ∧ v =
X
vl0
∧
l<m0
k=m
X
l<m0
t0l
+M
∞
X
λl < t0 ,
l=m0
with (u, s)Φk (vk , tk ), (u, s)Φl (vl0 , t0l ). W.l.o.g., we may assume m0 > m. Since we
both have (u, s)Φk (vk , tk ) and (u, s)Φk (vk0 , t0k ), we know that ||vk − vk0 || ≤ tk + t0k
for k < m. Since u ∈ K = dom(fk ) for m ≤ k < m0 , (u, s)Φk (vk0 , t0k ) implies
||λk f (u) − vk0 || ≤ t0k , so that ||vk0 || ≤ t0k + λk ||f (u)|| ≤ t0k + M λk . Putting things
together, we find
X
X
0
0 ||v − v || = vk −
vk 0
k<m
≤
X
k<m
X
||vk0 ||
m≤k<m0
k<m
≤
X
||vk − vk0 || +
tk + M
X
m≤k<m0
k<m
76
λk +
X
k<m0
t0k < t + t0 .
• Suppose (u, s)Φs (v, t) and (u, s)Φe (v 0 , t0 ). Then we have (u0 , s0 )Φs (v 0 , t0 ) such
that ||u − u0 || + s < s0 , and natural numbers m, m0 ∈ N+ , (vk , tk ) ∈ E × Q+ ,
∀k < m, (vl0 , t0l ) ∈ E × Q+ , ∀l < m0 , such that
v=
X
vk ∧
k<m
X
tk + M
k<m
∞
X
0
λk < t ∧ v =
X
vl0
∧
l<m0
k=m
X
t0l
+M
l<m0
∞
X
λl < t0 ,
l=m0
with (u, s)Φk (vk , tk ), (u0 , s0 )Φl (vl0 , t0l ). Since then also (u, s)Φl (vl0 , t0l ) for l < m0 ,
we are able to find ||v − v 0 || < t + t0 in an analogous way as before.
• Suppose (u, s)Φe (v, t) and (u, s)Φe (v 0 , t0 ). Then we have (u0 , s0 )Φs (v, t) and
(ũ, s̃)Φs (v 0 , t0 ) such that ||u − u0 || + s < s0 and ||u − ũ|| + s < s̃, and natural
numbers m, m0 ∈ N+ , (vk , tk ) ∈ E × Q+ , ∀k < m, (vl0 , t0l ) ∈ E × Q+ , ∀l < m0 ,
such that
v=
X
k<m
vk ∧
X
tk + M
k<m
∞
X
0
λk < t ∧ v =
X
vl0
l<m0
k=m
∧
X
t0l
+M
l<m0
∞
X
λl < t0 ,
l=m0
with (u0 , s0 )Φk (vk , tk ), (ũ, s̃)Φl (vl0 , t0l ). Since then also (u, s)Φk (vk , tk ) for k < m,
and (u, s)Φl (vl0 , t0l ) for l < m0 , we are able to find ||v −v 0 || < t+t0 in an analogous
way as before.
2. Suppose (u, s)Φ(v, t) and ||u − u0 || + s0 < s for (u0 , s0 ) ∈ E × Q+ .
• If (u, s)Φs (v, t), then (u0 , s0 )Φe (v, t), so that (u0 , s0 )Φ(v, t).
• If (u, s)Φe (v, t), then we have (ũ, s̃)Φs (v, t) such that ||u − ũ|| + s < s̃. Then
||u0 − ũ|| + s0 ≤ ||u0 − u|| + ||u − ũ|| + s0 < s − s0 + s̃ − s + s0 = s̃ ,
so that (u0 , s0 )Φe (v, t) and again (u0 , s0 )Φ(v, t).
3. Suppose (u, s)Φ(v, t) and ||v − v 0 || + t < t0 for (v 0 , t0 ) ∈ E × Q+ .
• If (u, s)Φs (v, t), then there exists m ∈ N+ , (vk , tk ) ∈ E × Q+ , ∀k < m, such
that
v=
X
vk ∧
k<m
X
k<m
tk + M
∞
X
k=m
with (u, s)Φk (vk , tk ). Defining
vk0 :=
v0 − v
+ vk ,
m−1
t0k :=
t0 − t
+ tk ,
m−1
77
λk < t ,
for k < m, we have
X
vk0 = v 0 − v +
k<m
X
k<m
t0k
+M
∞
X
X
vk = v 0 − v + v = v 0 ,
k<m
0
λk = t − t +
k=m
X
tk + M
k<m
∞
X
λk < t0 − t + t = t0 .
k=m
Moreover, we have for k < m
||vk − vk0 || + tk =
t0 − t
||v − v 0 ||
+ tk <
+ tk = t0k ,
m−1
m−1
so that (u, s)Φk (vk0 , t0k ). Hence, we have (u, s)Φs (v 0 , t0 ), so that (u, s)Φ(v 0 , t0 ).
• If (u, s)Φe (v, t), then we have (u0 , s0 )Φs (v, t) such that ||u − u0 || + s < s0 . As
before, we then also have (u0 , s0 )Φs (v 0 , t0 ). Hence, we have (u, s)Φe (v 0 , t0 ), so
that (u, s)Φ(v 0 , t0 ).
We will now show that the function φ coded by Φ is indeed f by proving the following
two properties:
1. dom(φ) = K.
Suppose x ∈ K and > 0. Let m ∈ N+ be such that
∞
X
k=0
λk −
∞
X
λk =
k=m−1
∞
X
k=m
λk <
,
3M
and define 0 := /(3(m − 1)). Since x ∈ dom(λ0 f0 ), there exists (u, s)Φ0 (v0 , t0 ) with
||u − x|| < s and t0 < 0 . Let q be any rational such that 0 < q < (s − ||u − x||)/2.
◦
As in the proof of Lemma B.1, we can find u0 ∈ E ∩ K, such that ||x − u0 || < q. We
then have
||u − u0 || + q ≤ ||u − x|| + ||x − u0 || + q < ||u − x|| + 2q < s ,
so that (u0 , q)Φ0 (v0 , t0 ) with ||u0 −a|| < r, ||x−u0 || < q and t0 < 0 . In general, suppose
for k = 0, . . . , l < m − 1, we have (u, s)Φk (vk , tk ), with ||u − a|| < r, ||x − u|| < s
and tk < 0 . Since x ∈ dom(λl+1 fl+1 ), we find (u0 , s0 )Φl+1 (vl , tl ) with ||x − u0 || < s0
and tl < 0 . Let q be any rational such that
s − ||u − x|| s0 − ||u0 − x||
0 < q < min
,
,
2
2
◦
and ũ ∈ E ∩ K such that ||x − ũ|| < q. Then
||u − ũ|| + q ≤ ||u − x|| + ||x − ũ|| + q < ||u − x|| + 2q < s ,
||u0 − ũ|| + q ≤ ||u0 − x|| + ||x − ũ|| + q < ||u0 − x|| + 2q < s0 ,
78
so that (ũ, q)Φk (vk , tk ) for k = 0, . . . , l + 1, with ||ũ − a|| < r, ||x − ũ|| < q and tk < 0 .
Hence, we may conclude there exists m ∈ N, (u, s), (vk , tk ) ∈ E × Q+ , ∀k < m, with
||u − a|| < r and tk < 0 . Moreover, we have
X
∞
X
tk + M
k<m
λk < (m − 1)0 +
k=m
2
=
.
3
3
Now let v := v0 + . . . + vk and t be any rational such that 2/3 < t < . Then we
have (u, s)Φs (v, t), so that (u, s)Φ(v, t).
Conversely, suppose x ∈ dom(φ), i.e., for every > 0, we have (u, s)Φ(v, t) with
||u − x|| < s and t < . If (u, s)Φe (v, t), then we have (u0 , s0 )Φs (v, t) such that
||u − u0 || + s < s0 , and
||x − u0 || ≤ ||x − u|| + ||u − u0 || < s + s0 − s = s0 ,
so that cleary we may assume that for every > 0, we have (u, s)Φs (v, t) with
◦
||u − x|| < s and t < . This means that for every > 0, there exists some u ∈ K ∩ E
with ||u − x|| < s, m ∈ N+ and (vk , tk ) ∈ E × Q+ , ∀k < m, such that
X
(u, s)Φk (vk , tk ) ∧
∞
X
tk + M
k<m
λk < t < .
k=m
In particular, we find for any > 0 some (m, u, s, v0 , t0 ) ∈ Φ0 , such that
||u − x|| < s ∧ t0 < ,
so that x ∈ dom(f0 ) = K.
2. φ(x) = f (x) for all x ∈ dom(φ).
Suppose x ∈ dom(φ) = K. We need to show that φ(x) = f (x). Hence, let (u, s)Φ(v, t)
and ||x − u|| < s. We need to show that ||f (x) − v|| ≤ t.
• If (u, s)Φs (v, t), then there exists m ∈ N+ , (vk , tk ) ∈ E × Q+ , ∀k < m, such
that
∞
X
X
X
v=
vk ∧
tk + M
λk < t ,
k<m
k<m
k=m
with (u, s)Φk (vk , tk ). Then
∞
X
X
||f (x) − v|| = λk fk (x) −
vk k=0
≤
X
k<m
||λk fk (x) − vk || +
k<m
≤
X
∞
X
k=m
tk + M
k<m
∞
X
k=m
79
λk < t .
λk ||fk (x)||
• If (u, s)Φe (v, t), then we have (u0 , s0 )Φs (v, t) such that ||u − u0 || + s < s0 . Since
||x − u0 || ≤ ||x − u|| + ||u − u0 || < s + ||u − u0 || < s0 ,
we again have ||f (x) − v|| < t.
Hence, we conclude that Φ is a code for the continuous function f .
Theorem B.6. The following is provable in RCA0 . Let n ∈ N and Φn be as in Theorem
4.36, then Φn defines a code for Fn as a continuous function.
Proof. We will first check that Φn indeed codes a continuous function by checking the
properties of Definition 2.15. Note that we have
(u, s)Φn (v, t) ↔ ϕs (n, u, s, v, t) ∨ ϕe (n, u, s, v, t) ∨ ϕo (n, u, s, v, t) .
1. Suppose (u, s)Φn (v, t) and (u, s)n Φ(v 0 , t0 ).
• Suppose (u, s)Φn,s (v, t) and (u, s)Φn,e (v 0 , t0 ). Then it is easy to see that v = v 0 ,
so that ||v − v 0 || ≤ t + t0 .
• Suppose (u, s)Φn,s (v, t) and (u, s)Φn,e (v 0 , t0 ). Then we have (u0 , s0 )Φn,s (v 0 , t0 )
such that ||u − u0 || + s < s0 . Then it is easy to see that
||v − v 0 || ≤ ||u − u0 || < s + s0 < t + t0 .
• Suppose (u, s)Φn,e (v, t) and (u, s)Φn,e (v 0 , t0 ). Then we have (u0 , s0 )Φn,s (v, t) and
(ũ, s̃)Φn,s (v 0 , t0 ) such that ||u − u0 || + s < s0 and ||u − ũ|| + s < s̃. Again, it is
easy to see that
||v − v 0 || ≤ ||u0 − ũ|| ≤ ||u0 − u|| + ||u − ũ|| < s0 − s + s̃ − s < t + t0 .
0 0
• Suppose (u, s)Φn,s
(v,
t)
and
(u,
s)Φ
(v
,
t
).
Then
we
have
or
(u,
s)Φ
ṽ,
t̃
n,o
n,s
or (u, s)Φn,e ṽ, t̃ such that ||v 0 − ṽ|| + t̃ ≤ t0 . From what we already know, we
have ||v − ṽ|| ≤ t + t̃, so that
||v − v 0 || ≤ ||v − ṽ|| + ||ṽ − v 0 || ≤ t + t̃ + t0 − t̃ = t + t0 .
0 0
• Suppose (u, s)Φn,e
(v, t) and (u, 0s)Φn,o (v , t ). 0Then we have or (u, s)Φn,s ṽ, t̃
or (u, s)Φn,e ṽ, t̃ such that ||v − ṽ|| + t̃ ≤ t . As we already know, we then
have ||v − ṽ|| ≤ t + t̃, so that again
||v − v 0 || ≤ ||v − ṽ|| + ||ṽ − v 0 || ≤ t + t̃ + t0 − t̃ = t + t0 .
• Suppose (u, s)Φn,o (v, t) and (u, s)Φn,o (v 0 , t0 ). Then we have or (u, s)Φn,s ṽ, t̃
or (u, s)Φn,e ṽ, t̃ such that ||v − ṽ|| + t̃ ≤ t, and either (u, s)Φn,s v̂, t̂ or
(u, s)Φn,e v̂, t̂ such that ||v 0 − v̂|| + t̂ ≤ t0 . As we already know, we then have
||ṽ − v̂|| ≤ t̃ + t̂, so that
||v − v 0 || ≤ ||v − ṽ|| + ||ṽ − v̂|| + ||v̂ − v 0 || ≤ t − t̃ + t̃ + t̂ + t0 − t̂ = t + t0 .
80
2. Suppose (u, s)Φn (v, t) and ||u − u0 || + s0 < s for (u0 , s0 ) ∈ E × Q+ .
• If (u, s)Φn,s (v, t), then (u0 , s0 )Φn,e (v, t), so that (u0 , s0 )Φn (v, t).
• If (u, s)Φn,e (v, t), then we have (ũ, s̃)Φn,s (v, t) such that ||u − ũ|| + s < s̃. Then
||u0 − ũ|| + s0 ≤ ||u0 − u|| + ||u − ũ|| + s0 < s − s0 + s̃ − s + s0 = s̃ ,
so that (u0 , s0 )Φn,e (v, t) and again (u0 , s0 )Φn (v, t).
• If (u, s)Φn,o (v, t), then we have or (u, s)Φn,s (v 0 , t0 ) or (u, s)Φn,e (v 0 , t0 ), such that
||v − v 0 || + t0 < t. From what we already know, we either have (u0 , s0 )Φn,s (v 0 , t0 )
or (u0 , s0 )Φn,e (v 0 , t0 ), so that we have (u0 , s0 )Φn,o (v, t) and hence, (u0 , s0 )Φn (v, t).
3. Suppose (u, s)Φn (v, t) and ||v − v 0 || + t < t0 for (v 0 , t0 ) ∈ E × Q+ .
• If (u, s)Φn,s (v, t) ∨ Φn,e (v, t), then (u, s)Φn,e (v 0 , t0 ), so that (u0 , s0 )Φn (v 0 , t0 ).
• If (u, s)Φn,o (v, t), then we have either (u, s)Φn,s (ṽ, t̃) or (u, s)Φn,e (ṽ, t̃), such
that ||v − ṽ|| + t̃ < t. Then
||v 0 − ṽ|| + t̃ ≤ ||v 0 − v|| + ||v − ṽ|| + t̃ < t0 − t + t − t̃ + t̃ = t0 ,
so that (u, s)Φn,o (v 0 , t0 ) and again (u, s)Φn (v 0 , t0 ).
We will now show that the function φn coded by Φn is indeed Fn by proving the
following two properties:
1. dom(φn ) = K.
Suppose x ∈ K and > 0. Let 0 < q be any rational such that q < /4. As in the
◦
proof of Lemma B.1, we can find u ∈ E ∩ K, such that ||x − u|| < q. It is easy to
see that there exists v ∈ A such that (u, q)Φn,s (v, 4q), and hence, (u, q)Φn (v, 4q).
Conversely, suppose x ∈ dom(φn ), i.e., for every > 0, we have (u, s)Φ(v, t) with
||u − x|| < s and t < . If (u, s)Φn,e (v, t), then we have (u0 , s0 )Φn,s (v, t) such that
||u − u0 || + s < s0 , and
||x − u0 || ≤ ||x − u|| + ||u − u0 || < s + s0 − s = s0 .
If (u, s)Φn,o (v, t), then we either have (u, s)Φn,s (v 0 , t0 ) or (u, s)Φn,e (v 0 , t0 ) such that
||v − v 0 || + t0 < t. Then clearly also t0 < . Hence, we may assume that for every
> 0, we have (u, s)Φs (v, t) with ||u − x|| < s and s = t/4 < /4. But this means
◦
that for every > 0, there exists u ∈ A ∩ K with ||u − x|| < , which is only possible
if x ∈ K.
2. φn (x) = Fn (x) for all x ∈ dom(φ).
Suppose x = hxk : k ∈ N, xn ∈ Ai ∈ dom(φ) = K. We need to show that
φn (x) = Fn (x). Hence, let (u, s)Φn (v, t) and ||x − u|| < s. We need to show that
limk ||Fn (xk ) − v|| ≤ t, where for k ∈ N

 a0 , . . . an , xk(n+1) , . . . , xk(lh(xk )−1) if lh(xk ) > n + 1
Fn (xk ) =
.
α
if lh(xk ) ≤ n + 1
n
81
Note that ||Fn (xk ) − Fn (xl )|| ≤ ||xk − xl || for each k, l ∈ N, so that the sequence
hFn (xk ) : k ∈ Ni indeed converges to a point in l2 .
• If (u, s)Φn,s (v, t), then
v=

 p0k , . . . , pnk , un+1 , . . . , ulh(u)−1 if lh(u) − 1 > n
hp0k , . . . , pnk i
if lh(u) − 1 ≤ n
,
where k is the minimal natural number such that
s
,
2−k ≤ √
n+1
and am = hpml : l ∈ Ni. Since
||xk − u|| ≤ ||x − u|| + ||x − xk || ≤ s + 2−k ,
we find
lim ||Fn (xl ) − v|| ≤ lim ||Fn (xl ) − Fn (xk )|| + ||Fn (xk ) − v||
l
l
v
u n
uX
≤ lim ||xl − xk || + t
|am − pmk |2 + ||xk − u||2
l
m=0
p
(n + 1)2−2k + s2 + 2s · 2−k + 2−2k
s
s
s2
2s2
≤√
+ 2s2 + √
+
n+1
n+1 n+1
√
≤ s + 5s
≤ 2−k +
< 4s = t .
• If (u, s)Φn,e (v, t), then we have (u0 , s0 )Φs (v, t) such that ||u − u0 || + s < s0 . Since
||x − u0 || ≤ ||x − u|| + ||u − u0 || < s + ||u − u0 || < s0 ,
we again have ||Fn (x) − v|| ≤ t.
• If (u, s)Φn,o (v, t), then we have (u, s)Φs (v 0 , t0 ) such that ||v − v 0 || + t0 < t. As
already known, we then have ||Fn (x) − v 0 || ≤ t0 . Moreover, we find
||Fn (x) − v|| ≤ ||Fn (x) − v 0 || + ||v 0 − v|| < t0 + t − t0 = t .
Hence, we conclude that Φn is a code for the continuous function Fn .
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