Bishop’s Lemma for Lipschitz Algebras
by
Ryan M. Northrup
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Clarkson University
Bishop’s Lemma for Lipschitz Algebras
A Thesis Proposal by
Ryan M. Northrup
Division of Mathematics and Computer Science
Mentor: Aaron B. Luttman, Ph.D.
March 12, 2010
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Contents
1 Introduction
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2 Real-valued Lipschitz Functions and Preliminary Results
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3 Metric Spaces and Generalized Lipschitz Functions
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4 Lipschitz Algebras, Subalgebras and Preliminary Results
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5 Bishop’s Lemma for Lipschitz Algebras
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6 Literature Review
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7 Methodology and Timetable for Future Research
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Abstract
Continuous and differentiable functions are two types of functions that are used
in calculus. A third type, called Lipschitz continuous functions, has properties
common to both continuity and differentiability. Lipschitz continuity is stronger
than continuity but weaker than differentiability. Continuous functions, differentiable functions, and Lipschitz functions each have individual properties, which can
be explored through the methods of calculus. Collections of functions, known as
function algebras, however, are less popular and less studied. A theorem, called
Bishop’s Lemma, guarantees the existence of functions that play a role similar to
multiplicative inverses. More study of Bishop’s Lemma is required to fully understand it, but it is currently known for algebras of continuous functions. This thesis
attempts to extend Bishop’s Lemma to general Lipschitz algebras.
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1
Introduction
Continuous functions are well known to students who have taken calculus. In calculus,
students learn vital theorems, such as the Extreme Value Theorem (EVT) and the Mean
Value Theorem (MVT) which describe properties of continuous and differentiable functions. Introductary calculus focuses on continuity and differentiability, but in-between
lies another property, Lipschitz continuity. Lipschitz continuity is the halfway mark: it
is stronger than continuity, yet it is weaker than differentiability. It is possible to extend theorems that are applicable to continuous and differentiable functions to Lipschitz
functions as well.
Properties of individual Lipschitz functions have been studied and are well-understood.
Properties of collections of functions, on the other hand, are still relatively unexplored.
We know more about collections of continuous functions than collections of Lipschitz
functions. Further research is required, and many interesting theorems of collections of
continuous functions are yet to be extended to collections of Lipschitz functions.
A collection of functions is called an algebra if it is closed under addition, multiplication, and scalar multiplication. Algebras of Lipschitz functions, also called Lipschitz
algebras, have properties due to both algebraic and analytic structure. There exists a
particular algebraic theorem whose analytic conclusion is similar to invertibility. The
theorem, called Bishop’s Lemma, is the cornerstone theorem of this work, and it is describe later on. The theorem has been proven in context of continuous functions, and it
holds for the collection of all Lipschitz functions, also called the full Lipschitz algebra.
Though a proof for the full algebra exists, such a result is misleading. The full algebra is
a case worthy of study, but the theorem needs to be proven for general Lipschitz algebras.
Through algebraic and analytic techniques, we devise a strategy to prove Bishop’s Lemma
for a stricter collection of continuous functions–a general algebra of Lipschitz functions.
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Real-valued Lipschitz Functions and Preliminary
Results
As was noted, Lipschitz functions are continuous functions. Continuous functions, however, are not necessarily Lipschitz. Similarly, differentiable functions are Lipschitz as will
be shown, but Lipschitz functions are not necessarily differentiable. Continuous functions with corners, like f (x)
p = |x| over R, are not differentiable, but they are Lipschitz.
Another function, f (x) = |x| over R is continuous, but not Lipschitz.
More concretely, Lipschitz continuity allows the corner pathology, but restricts cusps.
Figure 1 shows the relevant hierarchy of functions. The three functions in Figure 2 show
a function without corners or cusps, a function with a corner, and a function with a cusp.
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Figure 1: Each region presents a different set of functions: region a represents all functions; region b represents all continuous functions; region c represents all Lipschitz continuous functions; region d represents all differentiable functions.
Definition 1 (Real-valued Lipschitz Functions). A real-valued, continuous function f
from A ⊂ R to B ⊂ R is Lipschitz continuous if
|f (p) − f (q)|
< ∞,
|p − q|
p,q∈A
L = sup
where L is called the Lipschitz constant of the function f .
The following preliminary result is a known. The proof below uses two well-known
theorems from calculus, and it forms the first preliminary result of this proposal.
Preliminary Result 1. A differentiable function f : R −→ R on a closed and bounded
domain D is Lipschitz continuous.
Proof. By the Mean Value Theorem, there exists ξ ∈ D such that ξ ∈ [a, b] ⊂ D and
(a)
f 0 (ξ) = f (b)−f
. By the Extreme Value Theorem, f 0 attains its maximum modulus on
b−a
D. Thus
|f (b) − f (a)|
|f 0 (ξ)| =
.
|b − a|
Also by the EVT, the derivative is bounded, and
|f (b) − f (a)|
< ∞.
|b − a|
a,b∈D
sup |f 0 (ξ)| = sup
ξ∈D
By Definition 1, f is Lipschitz continuous.
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Figure 2: Three real-valued functions: plot a represents arctan(x) a continuous, Lipschitz, and differentiable function; plot
p b represents |x|, a continuous, non-differentiable
Lipschitz function; plot c represents |x| a continuous, non-Lipschitz function.
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Metric Spaces and Generalized Lipschitz Functions
A Lipschitz function can have a domain more general than just the real line as previously
presented. However, the domain requires a notion of distance. Such a domain is called a
metric space.
Definition 2. A metric space is a set M with a function ρ : M × M −→ R such that
for all m1 , m2 , m3 ∈ M
1.
2.
3.
4.
ρ(m1 , m2 ) ≥ 0
ρ(m1 , m2 ) = 0 ⇐⇒ m1 = m2
ρ(m1 , m2 ) = ρ(m2 , m1 )
ρ(m1 , m2 ) + ρ(m2 , m3 ) ≥ ρ(m1 , m3 ).
The function ρ is called the metric.
Preliminary Result 2. Let the set be C and let the metric ρ : C × C −→ R, be given by
ρ(c1 , c2 ) = |c1 − c2 |. Thus, the first three properties of metric spaces hold trivially. The
triangle inequality holds by triangle inequality for absolute values.
Another hypothesis of the preliminary results in this work is based upon a property
called the diameter. The diameter of a metric space is the supremum of all distances in
the space, supxi ,xj ∈M ρ(xi , xj ). Our definitions, theorems and properties all depend on a
finite diameter metric spaces, ρ(xi , xj ) < ∞. We require finite diameter because removal
of this restriction would allow a third continuity pathology, whose properties fall outside
the scope of this proposal.
The domains of generic Lipschitz functions are the finite-diameter metric spaces outlined above.
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Definition 3. Let x1 , x2 be elements of a finite-diameter metric space X with metric
ρX : X × X −→ R. Then a continuous function f : X −→ C is Lipschitz continuous
if
|f (x1 ) − f (x2 )|
L = sup
< ∞.
(1)
ρX (x1 , x2 )
x1 ,x2 ∈X
If finite, the value of 1 is called the Lipschitz Constant of f .
The definition above is necessary to define more general Lipschitz algebras.
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Lipschitz Algebras, Subalgebras and Preliminary
Results
A Lipschitz algebra is a mathematical structure similar to a vector space. An algebra
has Lipschitz functions as elements and three closed operations: multiplication, addition,
and scalar multiplication. Multiplication, addition, and scalar multiplication of functions
follows from algebraic multiplications of complex numbers.
More results are presented below. These results are known, but the proofs are original.
The purpose of these results is to familiarize and examine Lipschitz functions.
Preliminary Result 3. Let Lip(X ) be the full Lipschitz Algebra with domain X , f, g ∈
Lip(X ) and a ∈ C. Now, af ∈ X , f g ∈ X and f + g ∈ X .
Proof. The proofs for addition and scalar multiplication are similar, and to save space we
give only the proof of products stated above here. Since f, g are Lipschitz, then there exist
Lf , Lg that are expressed as supremums in the form of Definition 1. Now, by continuity,
the product f g(x) = f (x)g(x). So,
sup
x,y∈X
|f (x)g(x) − f (y)g(y)|
|f g(x) − f g(y)|
= sup
.
ρ(x, y)
ρ(x, y)
x,y∈X
(2)
Now by adding zero, [f (x)g(y) − f (x)g(y)] and reshaping the numerator of the fraction,
we get
sup
x,y∈X
|f (x)| |g(x) − g(y)| + |g(y)| |f (x) − f (y)|
≤ Lg sup f (x) + Lf sup g(y)
ρ(x, y)
x∈X
y∈X
which is less than the right hand side of Equation 2 by the triangle inequality. So,
Lf g ≤ Lg supx∈X f (x) + Lf supy∈X g(y) < ∞. Thus, the product of Lipschitz functions is
Lipschitz. A similar proof exists in [2].
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Thus, products, sums and scalar multiples of Lipschitz functions are also Lipschitz
functions. Thus, it makes sense to define operations between Lipschitz functions. When
these properties are placed on a collection of Lipschitz functions over a finite-diameter
metric space, then we have a formal Lipschitz algebra.
As stated previously, the result which we wish to prove is known to be true for the
full Lipschitz algebra. Though this step is in the right direction, this truth is not the
most general case. More generally, we would like our result to hold in structured subsets
of the full Lipschitz algebra, notably also including the full algebra as well. Smaller
subsets, if maintaining the structure of the full Lipschitz algebra, are called subalgebras.
To illustrate the idea of a subalgebra, we provide the following result.
Preliminary Result 4. Let Lip0 (X) ⊂ Lip(X) be the set of all Lipschitz functions with
the form f (0) = 0C on the interval [0, 1]. Let g, h ∈ Lip0 (X). So, g + h ∈ Lip0 (X) since
(g + h)(0) = 0C = 0C + 0C = g(0) + h(0). Also, gh ∈ Lip0 (X) because (gh)(x) = 0C =
(0C )(0C ) = g(x)h(x). For scalar multiplication, let a ∈ C, thus ag(0) = 0C = a(0C ) =
a ∗ g(0). Thus the set is closed under all operations of the subalgebra. Also, multiplicative
and additive inverses exist. Thus, Lip0 (X) is called a subalgebra of Lip(X)
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Bishop’s Lemma for Lipschitz Algebras
Bishop’s Lemma is a theorem about collections of continuous functions, connecting analysis and algebra. A general proof of the theorem for continuous functions can be found
in [6]. The theorem is known in the setting of general continuous functions, and it is
known for the full Lipschitz algebra; however, Bishop’s Lemma is yet to be proven
for general Lipschitz algebras. The following is the proposed theorem, and it is the
main research problem to be addressed by this thesis.
Proposition 1 (Main Research Hypothesis). Let X be a finite-diameter metric space, A
a Lipschitz algebra, f ∈ A, and x, x0 ∈ X . Then there exists g ∈ A such that
|f (x)g(x)| < |f (x0 )g(x0 )|,
(3)
for all x 6= x0 .
The above result is known for some function algebras. This thesis attempts to extend
these known proofs of Bishop’s Lemma found in [1] and [6] to general Lipschitz algebras.
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Literature Review
The area of mathematics in which this thesis resides is a relatively new and unstudied
crossroads between algebra and analysis. There are fewer than 20 researchers in the world
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who actively research in this field. This unpopularity may be due to the limited nature
of primary sources. For example, primary sources on Lipschitz algebras are limited to
one paper, [1], and one key text [2]. More research is available on metric spaces, and
[3] and [4] are useful beginning texts. The largest research endeavor, however, is due to
the lack of research material on Bishop’s Lemma. The origin of Bishop’s Lemma is a
mystery, attributed to Bishop in [5], no actual proof by Bishop himself is known. The
small number of researchers and difficulty of available material, such as the paper [6] and
[7], make for a short reading list on the main research problem. Material, especially for
Bishop’s Lemma, simply doesn’t exist. Since there is little material available, the research
is novel and exciting. Charting this mathematical territory makes this thesis interesting
and worthwhile.
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Methodology and Timetable for Future Research
The methodology for this thesis involves intensive literature reviews: finding sources and
deciphering them. Unlike other areas of mathematical research, work with Lipschitz algebras has been slow. In [2], Weaver puts this bluntly saying there is no reason why “many
of the theorems proven in the past decade could not have been found twenty or thirty
years earlier.” Indeed, some of the best sources are aging mathematical texts, papers
and lecture notes. Tracking down dated and scarce sources is the first challenging step.
Online searches, interlibrary loans, library visits, and book purchases get the resources
to the right place. With resources in hand, reading and understanding mathematics is
the next step in the process. Mathematicians learn new things by first memorizing and
visualizing definitions, then applying these definitions to examples. In this study, we used
metric spaces and Lipschitz functions to get us up to speed.
After a foundation anchors the mathematical researcher, proving known theorems is
useful practice to fully understand the mathematics. Research intuition and questions
form while proving known results. Adaptations previous proofs are also useful. Our
previous results benefit us directly, providing proof strategies and useful background.
The planned research in the first half of the timetable below is structured around
definitions and known results. The second portion of the timetable focuses on extensions
of those known results.
My Spring 2010 semester at Northumbria University ends with my last exam on May
21, 2010, and my time abroad officially ends on June 16, 2010. I will be picking up full
time work on the thesis, starting July 1, 2010, at Clarkson University. The summer work
will be directed by Dr. Aaron Luttman.
• The first week of July will be devoted to proving the theorem for the full Lipschitz
algebra, which is a known result.
• The second week of July will be devoted to generalizing the finite-diameter assumptions to compactness.
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• The third week of July will be devoted to previous proofs of Bishop’s Lemma in
the continuous functions.
• The final week of July will be devoted to formulating Lipschitz subalgebras, and
trying to find a strategy for proving/disproving Bishop’s Lemma.
• The first half of the Fall 2010 semester will be devoted to actually proving Bishop’s
Lemma for subalgebras, building upon strategies discussed over the summer.
• The second half of the Fall 2010 semester will be devoted to formalizing the results
with writing.
• The Spring 2011 semester will be devoted to writing and perhaps publishing the
work.
This project requires abstract algebra and introductory analysis. Though the tools
of calculus are suited to determine analytic properties of individual functions, collective
structures are best characterized by the tools of abstract algebra.
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References
[1] Jimnez-Vargas A., Luttman A., and Villegas-Vallecillos M.. “Weakly Peripherally
Multiplicative Surjections of Pointed Lipschitz Algebras.” Rocky Mountain J. Math.
To appear.
[2] Weaver N.. Lipschitz Algebras. World Scientific. 1999.
[3] Munkres J. R.. Topology, Second Edition. Prentice Hall. 2000.
[4] Kaplansky I.. Set Theory and Metric Spaces. Allyn and Bacon, Inc. 1972.
[5] Browder A.. Introduction to Function Algebras. W. A. Benjamin Inc.. 1969.
[6] Lambert S., Luttman A., Tonev, T.. “Weakly Peripherally-Multiplicative Mappings
Between Uniform Algebras.” Contempory Mathematics, 2007, 435, pp. 265-281.
[7] Hatori O., Hino K., Miura T., Oka H.. “Peripherally monomial-preserving maps between uniform algebras.” Mediterr. J. Math, 2009, Vol. 6, pp. 47-60.
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