CHARACTERISTIC PROPERTIES OF THE GURARIY SPACE
V.P. FONF AND P. WOJTASZCZYK
A BSTRACT. The Gurariy space G is defined by the property that for every pair
of finite dimensional Banach spaces L ⊂ M every isometry T : L → G admits
an extension to an isomorphisms T̃ : M → G with kT kkT −1 k ≤ 1 + . We
investigate the question when we can take T̃ to be also an isometry (i.e. = 0).
We identify a natural class of pairs L ⊂ M such that the above property for
this class with = 0 characterise the Gurariy space among all separable Banach
spaces. We also show that the Gurariy space G is the only Lindenstrauss space
such that its finite dimensional smooth subspaces are dense in all subspaces.
We dedicate this paper to the memory of Joram Lindenstrauss, a fine man and a geat mathematician
whose ideas influnced our whole professional lives.
1. I NTRODUCTION
An infinite-dimensional Banach space X is called a Lindenstrauss space if X ∗
is isometric to L1 (µ). This notion was studied in many papers, and in particular it
was proved in [LaLi] that a separable X is a Lindenstrauss space iff X = cl ∪∞
n=1
n , n = 1, 2, ..., and {X } is an increasing sequence of subspaces.
Xn , Xn = l∞
n
A separable Banach space G is called a Gurariy space if given ε > 0 and an
isometric embedding T : L → G of a finite-dimensional normed space L into G,
for any finite-dimensional space M ⊃ L there is an extension T̃ : M → G with
||T̃ ||||T̃ −1 || ≤ 1 + ε. The first example of such space G was given by V.I. Gurariy
in [G]. Gurariy space turned out to be a fascinating object with many unexpected
properties. Already in [G] it was proved that G has the following property: if
L, M ⊂ G are isometric finite-dimensional subspaces of G and I : L → M is an
˜ I˜−1 || <
isometry then for any ε > 0 there is an extension I˜ : G → G with ||I||||
1 + ε.
It was proved in [LaLi] that a Gurariy space is a Lindenstrauss space. W. Lusky
proved in [Lu] that a Gurariy space is isometrically unique (see also [KS]). It
was proved in [W] that any separable Lindenstrauss space is isometric to a 1complemented subspace of the Gurariy space (see also [Lu1]). A non-separable
analogs of the Gurariy space are considered in [ACCGM, GK].
Both authors were partially supported by the Foundation for Polish Science. First named author was
partially supported by the Israel Science Foundation, Grant 209/09 and by European Grant Spade 2.
Second named author was partially supported by the Center for Advanced Studies in Mathematics
of the Ben-Gurion University of the Negev, the “HPC Infrastructure for Grand Challenges of Science and Engineeringâ Project, co-financed by the European Regional Development Fund under the
Innovative Economy Operational Programme” and Polish NCN grant DEC2011/03/B/ST1/04902.
The authors would like to express their gratitude to the anonymous referee for his/her penetrating
criticism which lead to a substantial revision of the paper.
1
2
V.P. FONF AND P. WOJTASZCZYK
The starting point of our investigation was the following question: for which
pairs L ⊂ M in the definition of the Gurariy space an extension T̃ may be
chosen to be an isometry?
Before we proceed let us recall some definitions we use in the paper. First
we remark that we deal with real spaces only. Following the standard usage for
a normed space X the symbol BX will denote the unit ball of X i.e. BX =
{x ∈ X : kxk ≤ 1} and the symbol SX will denote the unit sphere of X
i.e. SX = {x ∈ X : kxk = 1}. A point x ∈ SX of unite sphere SX of
a normed space X is called a smooth point of SX if there is unique functional
f ∈ SX ∗ with f (x) = 1. A Banach space is called smooth if any point x ∈ SX is a
smooth point. A Banach space X is called polyhedral if for any finite-dimensional
subspace L ⊂ X the unite ball BL is a polytope, i.e. BL is intersection of finite
number of closed half-spaces. Let L, M ⊂ X be subspaces of a normed space X.
Define
θ(L, M ) = max{sup{d(x, BM ) : x ∈ BL }, sup{d(x, BL ) : x ∈ BM }.
θ is a metric in the family = of all closed subspaces of a normed space X, and
if X is a Banach space then a metric space (=, θ) is complete.
The following definition will be very important in our investigation.
Definition 1.1. We say that the pair L ⊂ M of normed spaces has the unique
Hahn-Banach extension property (UHB in short) if for any functional f ∈ L∗
there is a unique extension fˆ ∈ M ∗ with ||fˆ|| = ||f ||.
Note that x ∈ SM is a smooth point of SM iff the pair L = [x] ⊂ M has UHB.
We can summarise the main results of our paper in the following two theorems.
Theorem 1.2. Let X be a separable Banach space. TFAE
(a) X = G.
(b) Let L ⊂ M, dimL < ∞, codimM L = 1, be a pair with property UHB and
let T : L → X be an isometric embedding of L into X. Then there is an isometric
extension T̃ : M → X of T .
Theorem 1.3. For a separable Lindenstrauss space X TFAE:
(i) X = G.
(ii) Let L1 and L2 be finite-dimensional subspaces of X both isometric to `n∞ such
that the pairs L1 ⊂ X and L2 ⊂ X has UHB, and let I : L1 → L2 be an isometry.
Then there is a rotation (isometry onto) ψ : X → X such that ψ|L1 = I.
(iii) The family of all smooth finite-dimensional subspaces of X is dense (in the
metric θ) in the family of all finite-dimensional subspaces of X.
Note that the equivalence of (i) and (ii) strengthens a result from [Lu] which
says that for any two smooth points x, y ∈ SG there is an isometry T : G → G
with T x = y.
Now we will present some results that will be useful for us later.
Lemma 1.4. Let X be a separable Lindenstrauss space. Assume that X = cl∪∞
n=1
Xn , where Xn is an increasing sequence of subspaces such that each Xn is ison . Then there is a sequence {e }∞ ⊂ extB ∗ with w ∗ −cl{±e }∞ ⊃
metric to l∞
i i=1
i i=1
X
extBX ∗ , and such that extBXn∗ = {±ei |Xn }ni=1 , n = 1, 2, ....
CHARACTERISTIC PROPERTIES OF THE GURARIY SPACE
3
In the proof of this Lemma we will use
Theorem 1.5 (Milman). Let X be a Banach space and K ⊂ X ∗ be a w∗ -compact
convex set. Let T ⊂ K be such that w∗ -clco T = K. Then ext K ⊂ w∗ -clT .
Proof. From the Krein-Milman theorem we easily obtain that for any n ∈ N there
are functionals {eni }ni=1 ⊂ extBX ∗ such that for any m ≤ n we have
∗
{±eni |Xm }m
i=1 = extBXm
∗ =
Let ei be a w∗ -limit point of the sequence {eni }∞
n=2 , i = 1, 2, .... Clearly, extBXm
∗
{±ei |Xm }m
,
m
=
1,
2,
....
To
check
that
each
e
belongs
to
extB
assume
that
i
X
i=1
ei = 12 (f + g), f, g ∈ BX ∗ , f 6= g. Since the union ∪∞
n=1 Xn of the increasing
sequence Xn is dense in X, it follows that there exists an integer m, m ≥ i, such
∗ , a contradiction. To prove that
that f |Xm 6= g|Xm . However ei |Xm ∈ extBXm
∗
w∗ − cl{±ei }∞
⊃
extB
,
just
note
that
the
set
{±ei }∞
X
i=1
i=1 is 1-norming, hence
∗
by the separation theorem we have w − cl co{±ei }∞
=
BX ∗ . Now the Lemma
i=1
follows from Theorem 1.5.
¿From Lemma 1.4 follows that for any n the restriction en+1 |Xn has a representation
n
n
X
X
en+1 |Xn =
ain ei |Xn ,
|ain | ≤ 1, n = 1, 2, ....
i=1
i=1
The matrix (ain )i≤n is called a representing matrix of X. This notion was introduced in [LaLi] by a different method (making use of so called “admissible bases”).
However for our purposes the definition above (which is equivalent to that given in
[LaLi]) is more convenient.
The following known properties of the Gurariy space (see [LaLi]) will be important for us:
Theorem 1.6. Let (ain )i≤n be a triangular matrix such that vectors (a1n , a2n , ..., ann , 0, 0, ...),
n = 1, 2, ..., are dense in the unit ball of `1 . Then the Lindenstrauss space with representing matrix (ain )i≤n is the Gurariy space.
Theorem 1.7. A separable Lindenstrauss space X is the Gurariy space iff
w∗ − cl extBX ∗ = BX ∗
(1.1)
We will also use the following easy
Fact 1.8. For a separable Banach space X the following are equivalent
(1) (1.1) holds
(2) for any finite-dimensional subspace E ⊂ X we have
{w∗ − cl extBX ∗ }|E = BE ∗
(3) if X = cl
∪∞
n=1
(1.2)
Xn , then
{w∗ − cl extBX ∗ }|Xn = BXn∗
(1.3)
for n = 1, 2, . . .
Proof. Implications (1.1) =⇒ (1.2) and (1.2) =⇒ (1.3) are straightforward,
so we will prove that (1.3) implies (1.1). Assume to the contrary that there is
f ∈ BX ∗ \ w∗ − cl extBX ∗ . Clearly there is an m and w∗ −neighborhood of f
W = {h ∈ BX ∗ : |h(xi ) − f (xi )| < ε, i = 1, ..., n, ε > 0}
4
V.P. FONF AND P. WOJTASZCZYK
with xi ∈ Xm for i = 1, 2, . . . , n such that W ∩ w∗ − cl extBX ∗ = ∅. Now an
easy argument shows that f |Xm 6∈ {w∗ − cl extBX ∗ }|Xm , contradicting (1.2).
We will also use the following Lemma 1.9
Lemma 1.9. For the Gurariy space G a sequence {ei } in Lemma 1.4 may be
chosen so that w∗ − cl{ei } = BG∗ .
Proof. We use notation of Lemma 1.4. Fix a sequence {εn } of positive numbers
tending to zero. By using Theorem 1.7 and the conclusion w∗ − cl{±ei }∞
i=1 ⊃
extBX ∗ of Lemma 1.4 we deduce that for any integer m we have w∗ −cl{±ei }∞
i=m ⊃
extBX ∗ . It follows that there are an integer p > 1 and a sign α = ± such
that ||(−e1 − αep )|X1 || < ε1 . We redefine ep to be αep and keep the signs
of all other ei ’s, i = 1, ..., p − 1. Next we find a finite sequence of integers
p < q1 < q2 < ... < qp and a finite sequence of signs {αi }pi=1 such that
||(−ei − αi eqi )|Xp || < ε2 , i = 1, ..., p. Now we redefine eqi to be αi eqi , i =
1, ..., p, and keep the signs of all other ei ’s i = 1, ..., qp . The further proof is clear.
We conclude the introduction by the brief description of the paper. In chapter 2
we collect auxiliary results on finite-dimensional polyhedral spaces. Chapter 3 is
devoted to the proof of Theorem 1.2 and some of its corollaries. In chapters 4 and
5 we prove Theorem 1.3.
2. S OME PROPERTIES OF POLYHEDRAL FINITE - DIMENSIONAL PAIRS WITH
UHB
In this section we prove some properties of pairs of finite dimensional polyhedral
spaces with UHB which will be used as auxiliary results in sections 3 and 4.
Let us recall the concept of the face of a a symmetric convex closed and bounded
body in P ⊂ Rn . We say that γ ⊂ P is a face of P if it is closed and for any x ∈ γ
if x = ay + (1 − a)z, with y, z ∈ γ it follows that y, z ∈ γ. Let Y be a finitedimensional polyhedral space and γ be a face of BY . We denote γ̃ the affine span
of γ. It is easy to see that intγ̃ γ 6= ∅. Moreover for any x ∈ γ there is a (unique)
face γx ⊂ γ of BY with x ∈ intγ˜x γx .
Proposition 2.1. Let L ⊂ M be a pair of polyhedral finite-dimensional spaces
with dim L = l < n = dim M. TFAE:
(1) The pair L ⊂ M has UHB.
(2) For any x ∈ SL the restriction map from VM (x) =: {f ∈ SM ∗ : f (x) = 1}
onto VL (x) =: {f ∈ SL∗ : f (x) = 1} is 1-1.
(3) For any face γ ⊂ BM with γ ∩ L 6= ∅. we have L + γ̃ = M.
Proof. It is easy to see that (2) is just a reformultion of (1) so (1)⇔(2).
(2)⇒(3). Let us fix a face γ ∈ BM and an x ∈ L ∩ γ. If γx ⊂ γ is a face of BM
such that x ∈ intγ̃ γ (as described above) then γ 0 = γx ∩ L is a face of BL with
x ∈ intγ̃ 0 γ 0 . A standard argument shows that dim VM (x) = n − dim γx , and it
easily follows from (2) that dim VL (x) = l − dim γ 0 , so n − dim γx = l − dim γ 0 .
Moreover dim(L + γx ) = l + dim γx − dim γ 0 so we get dim(L + γ̃) ≥ dim(L +
γx ) = n which gives the claim
(3)⇒(1). Let f ∈ L∗ and let F denotes the set of all norm preserving extensions
of f to M . Let
\
γ=
{x ∈ BM : g(x) = kf k}
g∈F
CHARACTERISTIC PROPERTIES OF THE GURARIY SPACE
5
Clearly it is a face of BM and γ ∩ L 6= ∅. Moreover each g ∈ F takes the value
kf k on γ̃. From (2) we infer that M = L + γ̃ so for any x = l + z ∈ M with l ∈ L
and z ∈ γ̃ and for any g ∈ F we have g(x) = g(l) + g(z) = f (l) + kf k. This
shows that f has a unique norm preserving extension to M .
Corollary 2.2. Let M be a finite-dimensional polyhedral space and E ⊂ M be a
hyperplane. If E does not contain any face of BM then the pair E ⊂ M has UHB.
Definition 2.3. Let E be a polyhedral finite-dimensional space and extBE ∗ =
n as
{±hi }ni=1 . We define a canonical embedding ψE : E → l∞
ψE x = (hi (x))ni=1 ,
x ∈ E.
n
We say that E is a fine space and BE is a fine polytope if the pair ψE (E) ⊂ l∞
has UHB.
Note that when we say that a polytop P is fine we mean that P is a unit ball of
a fine space.
n has UHB then E is fine. Indeed, let σ ⊂ {1, ..., n} be such
If the pair E ⊂ l∞
σ has UHB.
that {±ei |E}i∈σ = extBE ∗ . Then E ⊂ l∞
The following proposition gives an intrinsic characterization of fine spaces.
Proposition 2.4. Let E be a polyhedral finite-dimensional space. TFAE:
(a) E is fine.
(b) Any face of BE ∗ is a simplex.
(c) For any x ∈ extBE we have |{f ∈ extBE ∗ : f (x) = 1}| = dim E.
Proof. The equivalence (b) and (c) is clear. Assume that E is canonically embedn
n , i.e. if {±e }n
ded into l∞
i i=1 = extBl1n then extBE ∗ = {±ei |E}i=1 .
An easy consideration shows that for x ∈ SE
V`n∞ (x) = co{αi ei : αi ei (x) = 1, αi = ±1},
VE (x) = co{αi ei |E : αi ei (x) = 1, αi = ±1}.
This implies that the restriction map from V`n∞ (x) onto VE (x) is 1-1 so the Proposition follows from Proposition 2.1.
Lemma 2.5. Let E be a polyhedral finite-dimensional space and 0 < γ < 1. Then
there is a fine polytope P ⊂ E with (1 − γ)P ⊂ BE ⊂ (1 + γ)P.
n , E 6= ln . Let Q : ln → E be a projector and
Proof. We can assume that E ⊂ l∞
∞
∞
n
let r > 0 be a small number to be fixed later. We first find a fine subspace L ⊂ l∞
n
with θ(E, L) < r. For a subspace Y ⊂ l∞ denote AY the family of all faces of
n which do not intersect Y. Put
Bl∞
aY = min{||x − y|| : x ∈ Y, y ∈ F, F ∈ AY }.
n which are not in
Clearly, aY > 0. Next, let BY be the family of all faces of Bl∞
AY , i.e. F ∈ BY iff F ∩ Y 6= ∅. A proof of the following Fact is standard.
n . Assume that F ∈ B and Y + aff spanF 6= ln . Then for
Fact 2.6. Let Y ⊂ l∞
Y
∞
n with θ(Z, Y ) < ε and such that Z ∩ F = ∅.
any ε > 0 there is a subspace Z ⊂ l∞
6
V.P. FONF AND P. WOJTASZCZYK
Using Fact 2.6 finitely many times we obtain a fine subspace L (we use here
Proposition 2.1) such that θ(E, L) < r.
Let x ∈ SL and y ∈ SE be such that ||x − y|| < r. Then we have ||Qx − x|| =
||Q(x − y) + (y − x)|| ≤ (||Q|| + 1)r. Hence
1 − (kQk + 1)r ≤ ||Qx|| ≤ 1 + (kQk + 1)r,
for any x ∈ SL . If P = Q(BL ) then clearly (1 − (||Q|| + 1)r)BE ⊂ P ⊂
(1 + (||Q|| + 1)r)BE . This implies that for r chosen sufficiently small we get
(1 − γ)P ⊂ BE ⊂ (1 + γ)P. Finally from Proposition 2.4 follows that P is a fine
polytope which completes the proof.
Proposition 2.7. Let E be a finite-dimensional polyhedral space and ε > 0. Then
there exists a finite-dimensional polyhedral space M, M ⊃ E, such that the pair
E ⊂ M has UHB, and a fine subspace L ⊂ M with θ(E, L) < ε.
Proof. Let E1 and E2 be two copies of the space E. Put X = E1 ⊕∞ E2 and let
L ⊂ X be a subspace such that L ∩ E1 = {0} and θ(E1 , L) < α (α > 0 will be
specified later). Since L ∩ E1 = {0}, it follows that
δ = min{||x − y|| : x ∈ SL , y ∈ E1 } > 0.
Fix two numbers β and d with 0 < β < 1, 0 < d < δ, and put
V = co{(BE2 + βBE1 ), (dBE2 + BE1 )}.
Since d < δ we see that L ∩ V ⊂ intBL . By Lemma 2.5 (with appropriate γ > 0)
there exists a fine polytope P ⊂ L such that (L ∩ V ) ⊂ intP and P ⊂ intBL . Put
W = co{V ∪ P } and let M be a Banach space X with the unit ball W. It is not
difficult to see that W ⊂ BX and W ∩ L = P. Clearly, the pair E ⊂ M has UHB.
Finally, by taking β ∈ (0, 1) close to 1 we can get W as close to BX as we like,
and by taking α > 0 small enough we get θ(E, L) < ε. The proof is complete.
Proposition 2.8. Let L ⊂ M be a pair of finite-dimensional polyhedral spaces
with UHB. Then there exists a chain of spaces
L = L0 ⊂ L1 ⊂ L2 ⊂ ... ⊂ Lm−1 ⊂ Lm = M
(2.4)
such that for any k = 0, 1, ..., m−1, the pair Lk ⊂ Lk+1 has UHB and codimLk+1 Lk =
1.
Proof. We start with construction of Lm−1 . Denote E = L⊥ the annihilator of
L in M ∗ . Since the pair L ⊂ M has UHB it follows that E 6⊂ Kerx, for any
x ∈ extBM . Indeed, each x ∈ extBM generates a maximal face γ = {f ∈
SM ∗ : f (x) = 1} of BM ∗ which is a solid part of a hyperplane Γ = {f ∈
M ∗ : f (x) = 1}, i.e γ = BM ∗ ∩ Γ and intΓ γ 6= ∅. If it were E ⊂ Kerx,
for some x ∈ extBM , then for any f ∈ intΓ γ we would get that the intersection
(f + E) ∩ BM ∗ is not a singleton, contradicting UHB. Therefore E ∩ Kerx is a
hyperplane in E, for any x ∈ extBM . Since the set extBM is finite it follows that
there is h ∈ E \ ∪x∈extBM Kerx. Put Lm−1 = {y ∈ M : h(y) = 0}. Clearly,
Lm−1 ⊃ L. Next if N = span{h} then L⊥
m−1 = N. Assume that g ∈ SM ∗ is such
that ||g|Lm−1 || = 1. Then since N is not parallel to any maximal face of BM ∗ , it
follows that (g + N ) ∩ BM ∗ is a singleton, which proves that the pair Lm−1 ⊂ M
has UHB.
CHARACTERISTIC PROPERTIES OF THE GURARIY SPACE
7
Since the pair L ⊂ M has UHB it follows that the pair L ⊂ Lm−1 has UHB, and
we can repeat the procedure above for the pair L ⊂ Lm−1 . Proceeding in this way
we construct the whole chain (2.4). The proof is complete.
Let us introduce the definition of a protected polytope.
Definition 2.9. Let E ⊂ X be a finite-dimensional polyhedral subspace of a Banach space X. We say that BE is a protected polytope in BX if there is a finite set
{hj }m
j=1 ⊂ SX ∗ such that if Γj = {x ∈ X : hj (x) = 1}, and γj = Γj ∩ BX , then
SE ⊂ intSX ∪m
j=1 γj .
It is not difficult to see that if BE is a protected polytope in BX then there is
r = r(E, X) > 0 such that for any z ∈ SX with d(z, E) ≤ r we have z ∈
intSX ∪m
j=1 γj . Also one can note that if X is a finite dimensional polyhedral space
than BE is a protected polytope for each subspace E ⊂ X.
Proposition 2.10. Let M be an n-dimensional normed space and ε > 0. Then
there is a 2n-dimensional normed space Z such that
(i) M ⊂ Z.
(ii) There is a polyhedral subspace E ⊂ Z with θ(M, E) < ε.
(iii) There is a chain M = Y0 ⊂ Y1 ⊂ Y2 ⊂ ... ⊂ Yn−1 ⊂ Yn = Z, such that each
pair Yk−1 ⊂ Yk has UHB and codimYk Yk−1 = 1, k = 1, ..., n,
Proof. Let us start with the following standard Lemma whose proof is omitted.
Lemma 2.11. Let {ei }ni=1 ⊂ SM be a basis of M. Given a normed space Z
containing M, and ε > 0 there is δ > 0 such that for any set {ui }ni=1 ⊂ Z with
||ei − ui || < δ for i = 1, ..., n, we have θ(M, [ui ]ni=1 ) < ε.
We construct spaces Z and E inductively by constructing two chains of spaces
M = Y0 ⊂ Y1 ⊂ Y2 ⊂ ... ⊂ Yn−1 ⊂ Yn = Z and E1 ⊂ E2 ⊂ . . . ⊂ En = E.
First we construct Y1 and E1 as follows. We fix four positive numbers α, β, a, A,
α < β such that
a
β
<
< δ.
1+β
A
and a symmetric, closed polytope in V ⊂ M such that (1 + α)BM ⊂ V ⊂
(1 + β)BM . We define the space Y1 as M ⊕ R with unit ball
BY1 = (BM ⊕ R) ∩ VA
where VA = co{(0, A), (0, −A), (V, 0)}. Geometrically BY1 is a double pyramid
with base V ⊂ M and vertices (0, ±A) trimmed around the base by a cylinder
with base BM ⊂ V . Denote
S1 = SY1 ∩ (BM ⊕ R), S2 = SY1 \ S1 .
Aβ
Next from V ⊂ (1 + β)BM and a > 1+β
it follows that if u1 = (e1 , a) then
u1 /||u1 || ∈ intSY1 S1 . Put E1 = [u1 ]. Since V is a polytope, VA is also a polytope
∗
in M ⊕ R. So we see that there exists a finite set of functionals {hj }m
j=1 ⊂ SY1
such that
S1 ⊂ ∪m
j=1 {x ∈ BY1 : hj (x) = 1}.
This shows that BE1 is a protected polytope in BY1 . To conclude the first step we
note that from Aa < δ it follows that ||e1 − u1 || < δ.
8
V.P. FONF AND P. WOJTASZCZYK
Now we assume that we have already constructed spaces Em ⊂ Ym , m =
1, ..., k, k < n, such that BEm is a protected polytope in BYm and the pairs
Ym−1 ⊂ Ym has UHB and codimYm−1 Ym = 1. Denote r = r(Ek , Yk ) (see the
remark after the definition of a protected polytope). We construct Ek+1 and Yk+1
as follows. We fix four positive numbers α, β, a, A, α < β such that
β
ra
<
< rδ,
1+β
A
and a symmetric, closed polytope V ⊂ M such that (1+α)BM ⊂ V ⊂ (1+β)BM .
We define the space Yk+1 as Yk ⊕ R with unit ball
BYk+1 = (BYk ⊕ R) ∩ VA
where VA = co{(0, A), (0, −A), (V, 0)}. Clearly codimYk Yk+1 = 1. Denote
S1 = SYk+1 ∩ (BYk ⊕ R), S2 = SYk+1 \ S1 .
Aα
¿From (1 + α)BM ⊂ V it follows that if |λ| ≤ 1+α
and x ∈ SM then (x, λ) ∈
S2 . Thus there exists a positive function φ : SYk → R such that S2 = {(x, λ) :
x ∈ SYk and |λ| < φ(x)}. In particular, one easily sees that the pair Yk ⊂ Yk+1
has UHB. Put uk+1 = (ek+1 , a) and Ek+1 = [Ek , uk+1 ].
∗
Clearly there exists a finite set of functionals {gj }sj=1 ⊂ SYk+1
such that
S1 = ∪sj=1 {x ∈ SYk+1 : gj (x) = 1}
(2.5)
Since BEk is a protected polytope in BYk there exists a finite set of functionals
∗
{hj }m
j=1 ⊂ SYk such that
SEk ⊂ intSYk ∪m
j=1 {x ∈ SYk : hj (x) = 1} =: W.
(2.6)
∗
Let us define extensions h̃j ∈ SYk+1
of hj by the condition h̃j (0, 1) = 0, j =
1, 2, . . . , m. It is clear that
V =: ∪m
j=1 {z ∈ SYk+1 : h̃j (z) = 1}
contains {(x, λ) : x ∈ W : |λ| ≤ φ(x)}. Using this we obtain that
int (S1 ∪ V ) ⊃ intSYk+1 S1 ∪ {(x, λ) : x ∈ W and |λ| ≤ φ(x)}.
(2.7)
(2.8)
If x ∈ Ek+1 then x = y + λuk+1 = (y + λek+1 , λa) with y ∈ Ek . Assume now
that x ∈ SEk+1 . This clearly implies that ky + λek+1 k ≤ 1. Let us consider two
cases
Case 1. ky + λek+1 k < 1. In this case x ∈ intSYk+1 S1 .
β
Case 2. ky + λek+1 k = 1. From V ⊂ (1 + β)BM and 1+β
< ra
A , it follows that
|λ| < r. Since BEk is a protected polytope in BYk , it follows from the definition of
r that (y + λek+1 ) ∈ W , so x ∈ {(x, λ) : x ∈ W and |λ| ≤ φ(x)}.
¿From (2.8) we get SEk+1 ⊂ int (S1 ∪ V ) and from (2.5) and (2.7) we get
that BEk+1 is a protected polytope in BYk+1 . Finally from Aa < δ it follows that
||ek+1 − uk+1 || < δ and from Lemma 2.11 that θ(M, E) < . This completes the
proof.
CHARACTERISTIC PROPERTIES OF THE GURARIY SPACE
9
3. I SOMETRIC EXTENSIONS OF ISOMETRIC EMBEDDINGS INTO THE G URARIY
SPACE
The main result of this section is the following
Theorem 3.1. Let X be a separable Banach space. TFAE
(a) X = G.
(b) Let L ⊂ M, dimL < ∞, codimM L = 1, be a pair with property UHB and
let T : L → X be an isometric embedding of L into X. Then there is an isometric
extension T̃ : M → X of T.
Remark 3.2. The condition UHB in Theorem 3.1 is important. Indeed, let e1 , e2
(2)
(2)
be a natural basis of the space l1 . Take L = [e1 ] and M = l1 . Next pick a
smooth point u1 ∈ smSG and define T : L → G by T e1 = u1 . Clearly, T does not
have an isometric extension on M.
To prove Theorem 3.1 we will need some additional definitions and two Lemmas.
Assume that L ⊂ M is a pair with UHB such that dimM < ∞, codimM L = 1.
Let l = L⊥ ⊂ M ∗ and s ∈ SM ∗ be such that l = [s]. Let Ω : L∗ → M ∗ , Ωf =
fˆ, f ∈ L∗ , be the norm-preserving extension map. It is not difficult to see that
Ω is continuous and 1-1. Next we define a map h : BL∗ → SM ∗ as follows. For
any f ∈ intBL∗ the straight line fˆ + l intersects the unit sphere SM ∗ in 2 points
g + = fˆ + t+ s, t+ > 0, and g − = fˆ + t− s, t− < 0. We put
h(f ) = fˆ + t+ s, f ∈ intBL∗ ,
h− (f ) = fˆ + t− s, f ∈ intBL∗ .
For f ∈ SL∗ we put h(f ) = h− (f ) = fˆ. An easy verification shows that the maps
h and h− are continuous and 1-1. Next we define the following functions
ω(r) = sup{||Ω(f ) − Ω(g)|| : f, g ∈ BL∗ , ||f − g|| ≤ r}, r ≥ 0,
β(r) = sup{||h(f )−h(g)||, ||h− (f )−h− (g)|| : f, g ∈ BL∗ , ||f −g|| ≤ r}, r ≥ 0,
and
ν(r) = sup{||g1 − g2 || : g1 , g2 ∈ BM ∗ , g1 |N = g2 |N, ||g1 |N || ≥ 1 − r}.
It is not difficult to see that
lim ω(r) = 0, lim β(r) = 0, lim ν(r) = 0
r→0
r→0
r→0
(3.9)
Finally put
+
ˆ
BM
∗ = {f + ts ∈ BM ∗ : f ∈ BL∗ , t ≥ 0}.
Now we prove two lemmas and then pass to the proof of Theorem 3.1. Let
E ⊂ X be a subspace of a Banach space X, h ∈ SX ∗ , and r > 0. Denote
S(E, h, r) = {f ∈ BX ∗ : ||(f − h)|E|| < r}.
Lemma 3.3. Let L ⊂ M be a pair of finite-dimensional normed spaces with UHB
and codimM L = 1. Let r > 0 and f ∈ BM ∗ and g ∈ BL∗ be such that ||f |N −
g|| < r. Then there is an extension ĝ ∈ BM ∗ of g such that ||f − ĝ|| ≤ β(r).
10
V.P. FONF AND P. WOJTASZCZYK
Proof. If [h(f |N ), h− (f |N )] is a segment connecting the points h(f |N ) and
h− (f |N ) then f ∈ [h(f |N ), h− (f |N )]. Hence f = λh− (f |N )+(1−λ)h(f |N ), λ ∈
[0, 1]. Put ĝ = λh− (g) + (1 − λ)h(g). Then ĝ|N = g, ĝ ∈ BM ∗ , and we have
||f − ĝ|| = ||(λh− (f |N ) + (1 − λ)h(f |N )) − (λh− (g) + (1 − λ)h(g))|| ≤
λ||h− (f |N ) − h− (g)|| + (1 − λ)||h(f |N ) − h(g)|| ≤ β(r).
The proof is complete.
Lemma 3.4. Let X be a Banach space and N ⊂ X be an n-dimensional subspace
of X with basis {yi }ni=1 . Then given γ > 0 there is δ > 0 such that for any
{xi }ni=1 , N1 = [xi ]ni=1 , with ||xi − yi || < δ, i = 1, ..., n, the following properties
hold.
(a) There is an automorphism ψ : X → X with
||ψ−I|| < γ, ||ψ −1 −I|| < γ, ||ψ|| ≤ 1+γ, ||ψ −1 || < 1+γ, ψ(yi ) = xi , i = 1, ..., n
(3.10)
∗ , and E ⊂ X be a subspace of X such that
(b) Let λ, α ∈ (0, 1), {hi }m
⊂
S
X
i=1
∪m
i=1 S(E, hi , α)|N1 ⊃ λBN1∗
(3.11)
p
−1
∪m
i=1 S(E, hi , 2α)|N ⊃ λ (1 + γ) BN ∗
(3.12)
Then
(c) Let λ, α ∈ (0, 1), {hi }m
i=1 ⊂ SX ∗ , and E ⊂ X be a subspace of X such that
∪m
i=1 S(E, hi , α)|N ⊃ λBN ∗
(3.13)
Then
∪m
i=1 S(E, hi , 2α)|N1 ⊃ λ
p
(1 + γ)−1 BN1∗
(3.14)
Proof. It is clear that by taking δ > 0 small enough we can guarantee that the set
{xi }ni=1 is linearly independent, i.e. the set {xi }ni=1 is a basis of N1 = [xi ]ni=1 .
Let {fi }ni=1 ⊂ X ∗ be a biorthogonal system for {yi }ni=1 , and {gi }ni=1 ⊂ X ∗ be a
biorthogonal system for {xi }ni=1 .
Take ε > 0 (which we specify latter) and choose δ > 0 so small that
P
P
1 + ni=1 ||fi ||δ 1 + ni=1 ||gi ||δ
P
P
max{
,
} < 1 + ε,
1 − ni=1 ||fi ||δ 1 − ni=1 ||gi ||δ
and define
ψ(x) = x +
n
X
fi (x)(xi − yi ),
x ∈ X.
i=1
Clearly, ||ψ||
i ) = xi , i = 1, ..., n. If y =
Pn ≤ 1 + ε, ||ψ − I|| ≤ ε, ψ(yP
n
ψ(x) = x + P
f
(x)(x
−
y
),
then
x
=
y
−
i
i
i
i=1
i=1 fi (x)(xi − yi ), and hence
n
||x|| ≤ ||y|| + i=1 ||fi ||δ||x||. Finally we get
||x|| ≤
||ψ(x)||
Pn
≤ (1 + ε)||ψ(x)||,
1 − i=1 ||fi ||δ
i.e. ||ψ −1 || ≤ 1 + ε. Next
||ψ
−1
y − y|| = ||
n
X
i=1
fi (ψ
−1
y)(xi − yi )|| ≤
n
X
i=1
||fi ||δ||ψ −1 ||||y|| ≤ ε(1 + ε)||y||,
CHARACTERISTIC PROPERTIES OF THE GURARIY SPACE
11
i.e. ||ψ −1 − I|| ≤ ε(1 + ε). Clearly, by taking δ > 0 in the consideration above
small enough we can make ε > 0 so small as we like which finishes the proof of
(a)
To prove (b) put gi = ψ ∗−1 hi , i = 1, ..., m. Then we have ||hi −gi || < ε(1+ε).
By taking ε > 0 small enough we can assume that ||hi − gi || < α/2. An easy
verification shows that S(E, hi , α) ⊂ S(E, gi , 23 α). If F = ∪i S(E, hi , α) and
F̂ = ∪i S(E, gi , 23 α) then
F ⊂ F̂
(3.15)
ˆ
ˆ
¿From (3.11) follows that for any f ∈ BX ∗ with f |N1 ∈ λBN ∗ we have
1
(fˆ + N1⊥ ) ∩ F 6= ∅
(3.16)
Also it is not difficult to see that N ⊥ = ψ ∗ (N1⊥ ). Fix f ∈ λ(1 + ε)−1 BX ∗ . Then
fˆ = ψ ∗−1 f ∈ λBX ∗ , and by (3.16) and (3.15) there is g ∈ N1⊥ with fˆ + g ∈ F̂ . It
follows that there is gi with
3
||((fˆ + g) − gi )|E|| < α,
2
i.e.
3
||(ψ ∗−1 (f + ψ ∗ g) − hi )|E|| < α.
2
Hence
3
3
||((f + ψ ∗ g) − hi )|E|| < ||ψ ∗ || α < (1 + ε) α,
2
2
and by taking ε > 0 small enough we get
7
||((f + ψ ∗ g) − hi )|E|| < α.
4
⊥
∗
⊥
Since g ∈ N1 it follows that ψ g ∈ N . Therefore we conclude that for any f ∈
λ(1+ε)−1 BX ∗ there are i, 1 ≤ i ≤ m, and t = ψ ∗ (fˆ+g) = f +ψ ∗ g, ||t|| ≤ 1+ε,
with ||(t − hi )|E|| < 47 α and t|N = f |N.
It follows that φ = (1 + ε)−1 t ∈ BX ∗ and φ|N = (1 + ε)−1 f |N. By taking
ε > 0 small enough we can get (φ − hi )|E|| < 2α, i.e. φ ∈ ∪i S(E, hi , 2α).
Finally take l ∈ (1 + ε)−2 BN ∗ and let f ∈ X ∗ be a Hahn-Banach extension of
(1 + ε)l. Clearly, f ∈ λ(1 + ε)−1 BX ∗ and from the consideration above follows
that there is φ ∈ ∪i S(ui , 2α) with φ|N = l.
By choosing ε > 0 and δ > 0 small enough we easily satisfy (3.12). The proof
of (b) is complete.
To prove (c) we just interchange N and N1 . The proof of the lemma is complete.
Remark 3.5. If dim E < ∞ then the set S(E, h, α) is w∗ -open in BX ∗ .
Proof of Theorem 3.1. (a)⇒(b). Put N = T (L). Let G = cl ∪ Xn , Xn =
n , n = 1, 2, .... Fix a sequence {γ }, 0 < γ < 1, tending to 0 (we specify it
l∞
k
k
latter). By using (3.9) find a sequence rk > 0 such that
β(rk ) < γk ,
ν(rk ) < γk
(3.17)
Apply Lemma 3.4 for each k, and find an increasing sequence mk and a sequence
of automorphisms ψk : G → G with ψk (N ) ⊂ Xmk , and such that the properties
(3.10) are satisfied (indeed, under the notation of Lemma 3.4 we take {xki }ni=1 ⊂
12
V.P. FONF AND P. WOJTASZCZYK
Xmk with mk large enough to satisfy ||xi − yi || < δk ). By passing to ψk /||ψk || we
can assume that
||ψk || = 1, ||ψk−1 || ≤ 1 + γk
(3.18)
Put Nk = ψk (N ) ⊂ Xmk . From (3.18) we have
(1 + γk )−1 BN ∗ ⊂ (ψk |N )∗ (BNk∗ ) ⊂ BN ∗
(3.19)
Since γk → 0, it follows that θ − lim Nk = N.
Put φk = ψk |N, φk : N → Nk , k = 1, 2, ....
Fact 3.6. For any i ∈ N holds ||φ∗1 (ei |N1 ) − φ∗2 (ei |N2 )|| < 2γ1 .
Proof. From (3.10) follows that ||ψ1 − ψ2 || < 2γ1 . Therefore for any x ∈ BN we
have
2γ1 > |ei (φ1 x) − ei (φ2 x)| = |(φ∗1 (ei |N1 ))(x) − (φ∗2 (ei |N2 ))(x)|,
which finishes the proof.
We construct an increasing sequence of integers {nk } and extensions φ̂k : M →
Xnk ⊂ G of φk ’s such that the sequence φ̂k (x) converges for any x ∈ M. Then
we define T̃ (x) = limk φ̂k (x), x ∈ M.
We start with k = 1. Using Theorem 1.7 we conclude that cl{ei |N1 } = BN1∗ .
From (3.19) and Lemma 1.9 follows that
(1 + γ1 )−1 BN ∗ ⊂ cl{φ∗1 (ei |N1 )} ⊂ BN ∗
1
that {φ∗1 (ei |N1 )}ni=m
is an r1 -net in
1 +1
m2 we redefine n1 = m2 , if n1 > m2
Find n1 > m1 such
see 3.17)). If n1 <
So WLOG we can assume that n1 = m2 . Put
(3.20)
)−1 B
(1 + γ1
N ∗ (for r1
we redefine m2 = n1 .
h1i = h(φ∗1 (ei |N1 )), i = 1, ..., m2 .
From the first inequality in (3.17) and (3.20) it follows that
−2
2
BM ∗ ⊃ co{±h1i }m
i=1 ⊃ (1 + γ1 ) BM ∗
(3.21)
Define a map φ̂1 : M → Xm2 ⊂ G as follows
2
φ̂1 (x) = (h1i (x))m
i=1 ,
x ∈ M,
and put M1 = φ̂1 (M ). It is not difficult to see that φ̂1 is an extension of φ1 , and
hence M1 ⊃ N1 . Moreover, it follows from (3.21) that
(1 + γ1 )−2 ||x|| ≤ ||φ̂1 (x)|| ≤ ||x||, x ∈ M
(3.22)
From (3.21) and 0 < γ1 < 1 we get
||φˆ1 || ≤ 1,
||φˆ1
−1
|| ≤ 1 + 3γ1
(3.23)
Next we construct an extension φ̂2 of φ2 . First we note that from (3.22) follows
that
BM ∗ ⊃ φ̂∗1 (BM1∗ ) ⊃ (1 + γ1 )−2 BM ∗
(3.24)
+
−2 B ∗ . Therefore if
It follows that (φ̂∗1 (SM1∗ ) ∩ BM
∗ )|N ⊃ (1 + γ1 )
N
+
−3
A = {f ∈ φ̂∗1 (SM1∗ ) ∩ BM
BN ∗ },
∗ : f |N ∈ (1 + γ1 )
then
A|N = (1 + γ1 )−3 BN ∗ .
(3.25)
CHARACTERISTIC PROPERTIES OF THE GURARIY SPACE
13
By using 3.25) we get
∗−1
∗−1
−3
−4
∗
φ̂∗−1
1 (A)|N1 = φ1 (A|N ) ⊃ φ1 ((1 + γ1 ) BN ∗ ) ⊃ (1 + γ1 ) BN1 ,
i.e. if A1 = φ̂∗−1
1 (A) then
A1 |N1 ⊃ (1 + γ1 )−4 BN1∗
(3.26)
Clearly, A is closed and A ∩ Ω(BN ∗ ) = ∅. It follows by a compactness argument
that d = dist(A, Ω(BN ∗ )) > 0. Put α = min{d, γ1 /4} and let {fi }m
i=1 ⊂ A1 be
m
∗
such that ∪i=1 (fi + α intBM1 ) ⊃ A1 . From the construction is clear that for any
∗
f ∈ ∪m
i=1 (fi + α intBM1 ) we have
||f || ≥ 1 − γ1 /4,
+
φ̂∗1 f ∈ BM
∗
(3.27)
Let hi ∈ BG∗ be a Hahn-Banach extension of fi , i = 1, ..., m. From (3.26) follows
−4
∗
that ∪m
(b) (λ = (1 +
i=1 S(M1 , hi , α)|N1 ⊃ (1 + γ1 ) BN1 . Apply Lemma 3.4,
p
−4
m
γ1 ) , E = M1 ), and conclude that ∪i=1 S(M1 , hi , 2α)|N ⊃ (1 + γ1 )−1 (1 +
−4
γ
apply Lemma 3.4, (c) (N1 = N2 , E = M1 , λ =
p1 ) BN ∗ . Next we again
(1 + γ1 )−1 (1 + γ1 )−4 ), and conclude (taking into account that γ2 < γ1 < 1)
that
−5
∪m
i=1 S(M1 , hi , 4α)|N2 ⊃ (1 + γ1 ) BN2∗ ⊃ (1 − 5γ1 )BN2∗
(3.28)
Next denote
σ = {i ∈ N : ei ∈ ∪m
i=1 S(M1 , hi , 4α)}.
Using Theorem 1.7, Remark 3.5 and (3.28) we obtain
cl{ei |N2 }i∈σ ⊃ (1 − 5γ1 )BN2∗
(3.29)
Next by using (3.19) and (3.29) we get
1 − 5γ1
BN ∗ ⊂ cl{φ∗2 (ei |N2 )}i∈σ
1 + γ1
(3.30)
Find a finite subset σ1 ⊂ σ such that {φ∗2 (ei |N2 )}i∈σ1 is an r2 -net in {φ∗2 (ei |N2 )}i∈σ .
1
∗
From (3.30) follows that the set {φ∗2 (ei |N2 )}i∈σ1 is an r2 -net for the set 1−5γ
1+γ1 BN .
∗
−1
Again by using Theorem 1.7 and φ2 (BN2∗ ) ⊃ (1 + γ2 ) BN ∗ we can find a finite
subset σ2 ⊂ {max σ1 + 1, max σ1 + 2, ...} such that
{φ∗2 (ei |N2 )}i∈σ2 ⊂ (1 + γ2 )−1 BN ∗ \ (1 − 5γ1 )(1 + γ1 )−1 BN ∗
(3.31)
and the set {φ∗2 (ei |n2 )}i∈σ2 is an r2 -net in
(1 + γ2 )−1 BN ∗ \ (1 − 5γ1 )(1 + γ1 )−1 BN ∗ .
Put n2 = max σ2 . If n2 < m3 we redefine n2 = m3 , if n2 > m3 we redefine
m3 = n2 . So WLOG we can assume that n2 = m3 .
Next we define extensions h2i (on M ) of the functionals φ∗2 (ei |N2 ), i = 1, ..., m3 .
Put
C2 = {m2 + 1, ..., m3 } \ (σ1 ∪ σ2 ).
To extend φ∗2 (ei |N2 ), for i ∈ {1, ..., m2 }, we use Fact 3.6 and Lemma 3.3 in the
following way. By Fact 1 ||φ∗1 (ei |N1 ) − φ∗2 (ei |N2 )|| ≤ 2γ1 . Next apply Lemma
3.3 with f = φ̂∗1 (ei |M1 ) and g = φ∗2 (ei |N2 ). Since (φ̂∗1 (ei |M1 ))|N = φ∗1 (ei |N1 )
14
V.P. FONF AND P. WOJTASZCZYK
it follows that ||f |N − g|| ≤ 2γ1 . Therefore Lemma 3.3 gives us an extension
ĝ ∈ BM ∗ with ||ĝ − f || ≤ β(2γ1 ). Put h2i = ĝ. Clearly
||φ̂∗1 ei − h2i || ≤ β(2γ1 ), 1 ≤ i ≤ m2
(3.32)
For i ∈ σ1 ∪ σ2 we define
h2i = h(φ∗2 (ei |N2 )).
For i ∈ C2 we define h2i inductively by using a representing matrix (aij ) of the
space G. Let C2 = {i1 < i2 <, ..., < ip }. Clearly, i1 > m2 , ip < m3 . Define
h2i1
=
iX
1 −1
aii1 h2i ,
i=1
and if
{h2ik }q−1
k=1
are already defined then we put
iq −1
h2iq
=
X
aiiq h2i .
i=1
An easy verification shows that
From (3.17) it follows that
h2i |N
= φ∗2 (ei |N2 ), i = 1, ..., m3 .
BM ∗ ⊃ co{±h2i } ⊃ (1 + γ2 )−2 BM ∗
(3.33)
Define the map φ̂2 : M → Xm3 ⊂ G by
3
φ̂2 (x) = (h2i (x))m
i=1 ,
x ∈ M.
It is not difficult to see that φ̂2 is an extension of φ2 , and hence M2 = φ̂2 (M ) ⊃
N2 . Moreover, it follows from (3.33) that
(1 + γ2 )−2 ||x|| ≤ ||φ̂2 (x)|| ≤ ||x||, x ∈ M
(3.34)
¿From (3.34) and 0 < γ2 < 1 we get
||φˆ2 || ≤ 1,
||φˆ2
−1
|| ≤ 1 + 3γ2
(3.35)
Next we estimate the distance maxx∈BM ||φ̂1 (x) − φ̂2 (x)||. First note that
max ||φ̂1 (x) − φ̂2 (x)|| = max max |ei (φ̂1 (x)) − ei (φ̂2 (x))| =
x∈BM
x∈BM i∈N
max max |(φ̂∗1 (ei ))(x) − (φ̂∗2 (ei ))(x)| = max ||φ̂∗1 (ei ) − φ̂∗2 (ei )||.
i∈N x∈BM
i∈N
It follows from the construction that we need to take into account i ≤ m3 only.
We consider 4 cases.
Case 1. i ∈ {1, ..., m2 }. From (3.32) we get
||φ̂∗1 (ei ) − φ̂∗2 (ei )|| ≤ β(2γ1 )
Case 2. i ∈ σ1 . Then ||ei |M1 || > 1 − γ1 . ¿From (3.23) we get
1 − γ1
≤ ||φ̂∗1 (ei |M1 )|| ≤ 1
1 + 3γ1
Note that
φ̂∗1 (ei |M1 ) = Ω(φ∗1 (ei |N1 )) + λi s,
(3.36)
(3.37)
CHARACTERISTIC PROPERTIES OF THE GURARIY SPACE
15
and since i ∈ σ1 it follows that λi > 0. If
g=
φ̂∗1 (ei |M1 )
||φ̂∗1 (ei |M1 )||
then from (3.37) follows that
||g − φ̂∗1 (ei |M1 )|| ≤ 1 −
1 − γ1
< 4γ1 .
1 + 3γ1
Hence
||g|N − (φ̂∗1 (ei |M1 ))|N || < 4γ1 ,
and since (φ̂∗1 (ei |M1 ))|N = φ∗1 (ei |N1 ) and g = h(g|N ), (recall that λi > 0) it
follows that
||φ̂∗1 (ei |M1 ) − h((φ̂∗1 (ei |M1 ))|N )|| ≤
||φ̂∗1 (ei |M1 ) − g|| + ||g − h((φ̂∗1 (ei |M1 ))|N )|| =
||φ̂∗1 (ei |M1 ) − g|| + ||h(g|N ) − h((φ̂∗1 (ei |M1 ))|N )|| ≤ 4γ1 + β(4γ1 ),
i.e.
||φ̂∗1 (ei |M1 ) − h((φ̂∗1 (ei |M1 ))|N )|| ≤ 4γ1 + β(4γ1 ),
Finally we have (by using (3.38) and Fact 3.6)
(3.38)
||φ̂∗1 (ei ) − φ̂∗2 (ei )|| = ||φ̂∗1 (ei |M1 ) − h(φ∗2 (ei |N2 ))|| ≤
||φ̂∗1 (ei |M1 ) − h(φ∗1 (ei |N1 ))|| + ||h(φ∗1 (ei |N1 )) − h(φ∗2 (ei |N2 ))|| ≤
4γ1 + β(4γ1 ) + β(2γ1 ),
i.e.
||φ̂∗1 (ei ) − φ̂∗2 (ei )|| ≤ 7β(4γ1 )
(3.39)
Case 3. i ∈ σ2 . Then (see (3.31))
||φ∗2 (ei |N2 )|| ≥
1 − 5γ1
> 1 − 6γ1
1 + γ1
(3.40)
Hence we have
||Ω(φ∗2 (ei |N2 )) − h(φ∗2 (ei |N2 ))|| ≤ ν(6γ1 )
(3.41)
Next from Fact 3.6 and (3.40) follows that
||φ∗1 (ei |N1 )|| ≥ 1 − 6γ1 − 2γ1 = 1 − 8γ1
(3.42)
It follows from the definition of ν(r) that
||Ω(φ∗1 (ei |N1 )) − φ̂∗1 (ei |M1 ))|| ≤ ν(8γ1 )
(3.43)
Finally by using that for i ∈ σ2 we have φ̂∗2 (ei |M2 ) = h+ (φ∗2 (ei )), (3.41), (3.43),
and the triangle inequality we get
||φ̂∗1 (ei |M1 ) − φ̂∗2 (ei |M2 )|| ≤ ||φ̂∗1 (ei |M1 ) − Ω(φ∗1 (ei |N1 ))||+
||Ω(φ∗1 (ei |N1 )) − Ω(φ∗2 (ei |N2 )) + ||Ω(φ∗2 (ei |N2 )) − h(φ∗2 (ei |N2 )) ≤
ν(8γ1 ) + β(2γ1 ) + ν(6γ1 ) ≤ 2ν(8γ1 ) + β(2γ1 ),
i.e.
||φ̂∗1 (ei |M1 ) − φ̂∗2 (ei |M2 )|| ≤ 2ν(8γ1 ) + β(2γ1 )
(3.44)
16
V.P. FONF AND P. WOJTASZCZYK
Case 4. i ∈ C2 . It easy follows from the definition of h2i , i ∈ C2 , that for i ∈ C2
we have
||φ̂∗1 (ei ) − φ̂∗2 (ei )|| ≤ max{||φ̂∗1 (ei ) − φ̂∗2 (ei )|| : i ∈ B2 }.
Finally by using (3.39) and (3.44) we get
max ||φˆ2 (x) − φˆ1 (x)|| ≤ 8β(4γ1 ) + 2ν(8γ1 )
x∈BM
(3.45)
The further construction is clear. In this way we construct a sequence of maps
φˆk : M → G, k = 1, 2, ..., such that each φˆk is an extension of φk , and such that
max ||φ̂k+1 (x) − φ̂k (x)|| ≤ 8β(4γk ) + 2ν(8γk ),
x∈BM
for any k. If we choose γk ’s such that
X
(8β(4γk ) + 2ν(8γk )) < ∞,
k
then the sequence {φˆk } converges to, say T̃ : M → G. From the construction is
clear that T̃ is an isometric extension of T. The proof of (a)⇒(b) is complete.
Proof of (b)⇒(a). We first prove that (b) implies that X is a Lindenstrauss space
and then show (1.1). Application of Theorem 1.7 completes the proof. To prove
that X is a Lindenstrauss space it is enough (see [LaLi]) to show that for any
finite-dimensional subspace M ⊂ X and any ε > 0, there is a subspace N ⊂ X
n with
isometric l∞
max{d(x, N ) : x ∈ M } < ε
(3.46)
By using Proposition 2.10 and (b) find a finite-dimensional polyhedral space Y ⊂
X with θ(M, Y ) < ε/2. Next by using Propositions 2.7, 2.8, and (b) we find a fine
subspace L ⊂ X with θ(L, Y ) < ε/2. Clearly, θ(L, M ) < ε. Finally, by using
the definition of a fine space, Proposition 2.8, and (b) we find a subspace N ⊂ X
n with (3.46). So we proved that X is a Lindenstrauss space.
isometric l∞
Next we check that w∗ − cl extBX ∗ = BX ∗ . Since X is a separable Linn , n = 1, 2, .... Clearly,
denstrauss space we have X = cl ∪n Xn , Xn = l∞
the w∗ −topology on BX ∗ is defined by Xn ’s. From Fact 1.8 we see that it is
Denote
enough to prove that cl (extBX ∗ |Xn ) = BXn∗ , for any n = 1, 2,
P....
n
n . Let {e }n
n
∗
L = Xn P
= l∞
i i=1 be a natural basis of l1 = L and f =
i=1 ai ei ∈
n
n+1 containing L in such a way
intBL∗ ,
|a
|
<
1.
Let
M
⊃
L
stronger
be
l
∞
i=1 i
Pn
∗ = ln+1 then e
that if {ei }n+1
is
a
natural
basis
of
M
n+1 |L =
1
i=1 ai ei |L.
i=1
Clearly, the pair L ⊂ M has property (UHB). Let T : L → X be a natural (isometric) embedding L into X. By the condition (b) of the theorem there
is an isometric extension T̃ : M → X. By the Krein-Milman theorem there is
e ∈ extBX ∗ with T̃ ∗ e = en+1 . It is easily seen that e|L = f which proves that
cl(extBX ∗ )|L = BL∗ . The proof of Theorem 3.1 is complete.
The following corollary shows that for a polyhedral space M the condition
codimM L = 1 in Theorem 3.1 can be omitted.
CHARACTERISTIC PROPERTIES OF THE GURARIY SPACE
17
Corollary 3.7. Let L ⊂ M be a pair of finite-dimensional polyhedral spaces (i.e.
BM is a polytope) with UHB. Assume that T : L → G is an isometry. Then there
is an isometric extension T̃ : M → G of T.
Proof. Apply Proposition 2.8 and Theorem 3.1, (a)⇒(b) which finish the proof.
Corollary 3.8. Let u ∈ G, u 6= 0, and M be a 2-dimensional normed space. Then
there is v ∈ G such that [u, v] isometric M.
Proof. WLOG we can assume that ||u|| = 1. Let x ∈ SM be a smooth point of
the unit sphere SM , and L = [x]. It is easy to see that the pair L ⊂ M has (UHB).
Define T : L → G by T x = u. Clearly, T is an isometric embedding, and by
Theorem 3.1 there is an isometric extension T̃ : M → G. Take any y ∈ M such
that [x, y] = M, and put v = T̃ y. The proof is complete.
Corollary 3.9. extBG = ∅
2 . If
Proof. Let u ∈ SG and u1 , u2 be a standard basis of the space M = l∞
L = [u1 ] then the pair L ⊂ M has (UHB). If T : L → G is defined by T u1 = u,
then by Theorem 3.1 there is an isometric extension T̃ : M → G. In particular,
||T̃ (u1 ± u2 )|| = 1, which proves that u is not an extreme point of BG .
Corollary 3.10. Let Y be a separable smooth Banach space (say Y = l2 ) and
E ⊂ Y be a finite-dimensional subspace of Y. Assume that E ⊂ G. Then there is
a subspace Z ⊂ G isometric with Y such that Z ⊃ E.
Proof. Apply Theorem 3.1 infinitely many times.
4. ROTATIONS OF THE G URARIY SPACE
The main result of this section is the following
Theorem 4.1. For a separable Lindenstrauss space X TFAE:
(a) Let L1 and L2 be two subspaces of X both isometric to `n∞ such that the pairs
L1 ⊂ X and L2 ⊂ X have UHB, and let I : L1 → L2 be an isometry. Then there
is a rotation (isometry onto) ψ : X → X such that ψ|L1 = I.
(b) X = G.
To prove Theorem 4.1 we need the following two Propositions.
Proposition 4.2. Let L ⊂ M with M isometric to `q∞ and L isometric to `p∞ with
p < q. Assume also that {±ei }qi=1 = extBM ∗ and {±ei |L}pi=1 = extBL∗ . Then
(a) L ⊂ M has UHB iff for any i, p + 1 ≤ i ≤ q, we have ||ei |L|| < 1.
p
q
r , p < q < r.
(b) Let L ⊂ M ⊂ N be normed spaces isometric to l∞
, l∞
, l∞
Assume that L ⊂ N has UHB. Then there exists a subspace M1 such that L ⊂
M1 ⊂ N and the pair M1 ⊂ N has UHB. Moreover there is an isometry A :
M → M1 with A|L = IdL .
18
V.P. FONF AND P. WOJTASZCZYK
Proof.
P (a). If for any i, p + 1 ≤ i ≤ q, we have ||ei |L|| < 1 then for any
g = qi=1 ai ei ∈ SM ∗ with g|L ∈ SL∗ we have ai = 0, for all i = p + 1, ..., q.
It easily follows that L ⊂ M has UHB. Next assume that L ⊂ M has UHB
and prove that for any i, p + 1 ≤ i ≤ q, we have ||ei |L|| < 1. Assume to
the contrary
have ||ej |L|| = 1. Hence
P that for some
Pp j, p + 1 ≤ j ≤ q, weP
p
ej |L = pi=1 bi ei |L,
|b
|
=
1.
If
g
=
e
,
h
=
j
i=1 i
i=1 bi ei , then g, h ∈ SM ∗
and g|L = h|L ∈ SL∗ . However g 6= h, contradicting that L ⊂ M has UHB.
(b). Let extBN ∗ = {±ei }ri=1 be ordered in such a way that extBM ∗ = {±ei |M }qi=1
P
Pp
j
and extBL∗ = {±ei |L}pi=1 . By (a) we have ej |L = pi=1 aji ei |L,
i=1 |ai | <
1, j = p + 1, ..., r. Define a map A : M → N as follows: if x ∈ M then put
P
y = Ax such that ei (y) = ei (x), for i = 1, ..., q, and ei (y) = pi=1 aji ei (x), for
i = q + 1, ..., r. It is not difficult to see that A is an isometric embedding with
A|L = IdL , and that M1 = A(M ) ⊂ N has UHB (by (a)). The proof is complete.
Proposition 4.3. (i) Let X be a Lindenstrauss space, X = cl ∪n Xn with Xn =
n
lP
∞ and let {ei }i ⊂ extBX ∗ be as described in Lemma 1.4 and let en+1 |Xn =
n
Pi=1 ain ei |Xn for n = 1, 2, ...,. Let {εn } be a sequence of positive numbers with
εn < ∞. Then there is an increasing sequence {En } of subspaces of X such
that
Pn
n and e
(1) En is isometric l∞
n+1 |En = (1 − εn )
i=1 ain ei |En , n = 1, 2, ....
P∞
(2) θ(Ep , Xp ) < i=p+1 εi , p = 1, 2, .... In particular cl ∪n En = X.
(3) Each pair Ep ⊂ X has UHB.
p
. Then for any
(ii) Let X be a Lindenstrauss space and Y ⊂ X be isometric to l∞
p
ε > 0 there is a subspace Z ⊂ X isometric to l∞ such that the pair Z ⊂ X has
UHB, and θ(Y, Z) < ε.
Proof. We fix an integer p and for n ≥ p we define a sequence of p dimensional
subspaces of Xn
Xpn = {x ∈ Xn : ek+1 (x) = (1−εk+1 )
k
X
aik ei (x), k = p, . . . , n−1}. (4.47)
i=1
It is clear that
conditions
Xpp
= Xp . For each y ∈ Xp and n ≥ p we define yn ∈ Xpn to satisfy
ej (yn ) = ej (y) for j = 1, 2, . . . , p
ek+1 (yn ) = (1 − εk+1 )
k
X
aik ei (yn ) for k = p, . . . , n − 1
(4.48)
(4.49)
i=1
Since functionals (ej |Xn )nj=1 are linearly independent such yn exists and is unique.
Note that yp = y. Moreover
ej (yn+1 ) = ej (yn ) for j = 1, . . . , n.
(4.50)
First we prove by induction that kyn k = kyk for n ≥ p. Let us recall that for
x ∈ Xµ we have kxk = maxj=1,...,µ |ej (x)|. ¿From (4.49) and (4.50) we get
en+1 (yn+1 ) = (1 − εn+1 )
n
X
i=1
ain ei (yn )
CHARACTERISTIC PROPERTIES OF THE GURARIY SPACE
19
so using the inductive assumption we obtain
kyn+1 k =
max
j=1,...,p+1
|ej (yn+1 )| = max |ej (yn )| = kyk.
j=1,...,p
Now we claim that the sequence (yn )n≥p converges. For n = p, p + 1, . . . we use
(4.50) and (4.49) to get
kyn+1 − yn k =
sup
|ej (yn+1 − ej (yn )|
j=1,...,n+1
= |en+1 (yn+1 ) − en+1 (yn )|
n
n
X
X
= (1 − εn+1 )
ain ei (yn ) −
ain ei (yn )
i=1
i=1
!
n
X
= εn+1
ain ei (yn ) ≤ εn+1 kyn k = εn+1 kyk
i=1
P
for n ≥ p. Using the telescoping sum we get kyn+m − yn k ≤ kyk n+m
i=n+1 for
n ≥ p and m ≥ 1 which in particular shows that the sequence (yn )n≥p converges
and
∞
X
ky − yn k ≤ kyk
εi .
(4.51)
i=p+1
Define an operator Tp : Xp → X by the formula Tp y = limn yn . It is clear from
the construction that Tp is an isometry and from (4.51) we get
ky − Tp (y)k ≤ kyk
∞
X
εi
(4.52)
i=p+1
for any y ∈ X
p . Put Ep = Tp (Xp ), p = 1, 2, . . . . From (4.52) follows that
P
∞
θ(Xp , Ep ) ≤
i=p+1 εi , and from the construction it is easily seen that Ep ⊂
Ep+1 , p = 1, 2, . . . .
To prove that each pair Ep ⊂ X has UHB we first note that by (a), Proposition
4.2 each pair Ep ⊂ Ep+1 has UHB. Next let g, h ∈ X ∗ be such that ||g|Ep || =
||h|Ep || = 1 and g|Ep = h|Ep . Since Ep ⊂ Ep+1 has UHB, it follows that
g|Ep+1 = h|Ep+1 , and proceeding in this way we get g|En = h|En , for any n.
Therefore g = h, and the proof of (i) is complete.
Claim (ii) immediately follows from (i) and a result from [LaLi] which says that
p
of a separable Lindenstrauss space X initiates a seany subspace Y isometric l∞
p+k
quence Y ⊂ Xp+1 ⊂ Xp+2 ⊂ ... such that each Xp+k is isometric to l∞
and
X = cl ∪∞
X
.
p+k
k=1
Proof of Theorem 4.1. (a)⇒(b). We will prove (1.3). We will assume that X =
cl ∪n Xn and those spaces satisfy (i) of Proposition 4.3. By Lemma 1.4 there is
n
a sequence {ei }∞
i=1 ⊂ extBX ∗ such that {±ei |Xn }i=1 = extBXn∗ , for any n.
q
Fix an integer p and ε > 0. Let {fi }i=1 be a finite ε-net in (1 − ε)BXp∗ . Clearly,
P
P
fi = pj=1 aij ei and pj=1 |aij | ≤ 1 − ε. Let
Y = {x ∈ Xp+q : ep+i (x) =
p
X
aij ei (x) for i = 1, ..., q}.
j=1
p
l∞
.
It is a subspace of Xp+q isometric to
From Proposition 4.2(a) follows that
Y ⊂ Xp+q has UHB. Since Xp+q ⊂ X has UHB, it follows that Y ⊂ X has UHB.
20
V.P. FONF AND P. WOJTASZCZYK
Let I : Xp → Y be a natural isometry of Xp onto Y, i.e. for x ∈ Xp
ei (Ix) = ei (x) for i = 1, ..., p and ei (Ix) = fi (x) for i = p + 1, ..., p + q.
By the condition (a) of the theorem there is a rotation T : X → X such that
T |Xp = I. Since T ∗ is a rotation of X ∗ it follows that T ∗ (extBX ∗ ) = extBX ∗ .
In particular, {T ∗ ep+i }qi=1 ⊂ extBX ∗ . However, (T ∗ ep+i )|Xp = fi , i = 1, ..., q.
Indeed, if x ∈ X then (T ∗ ep+i )(x) = ep+i (T x) = ep+i (Ix) = fi (x). It follows
that extBX ∗ |Xp is an ε-net in (1 − ε)BXn∗ . Since ε > 0 is arbitrary, it follows that
extBX ∗ |Xp is dense in BXn∗ which finishes the proof of (a)⇒(b).
(b)⇒(a). We will prove the following
Lemma 4.4. Let L1 , L2 ⊂ G be two subspaces isometric to `n∞ such that pairs
L1 ⊂ G and L2 ⊂ G have UHB. Let us fix an isometry I : L1 → L2 , a vector
x ∈ SG and > 0. Then there exist subspaces V1 , V2 ⊂ G isometric to `n∞1 such
that
(1) pairs V1 ⊂ G and V2 ⊂ G have UHB
(2) L1 ⊂ V1 and L2 ⊂ V2
(3) d(x, V1 ) < and d(x, V2 ) < (4) there exists an isometry ψ : V1 → V2 such that ψ|L1 = I.
The implication follows by inductively applying Lemma 4.4 for sequence {xi }∞
i=1
dense in SG and sequence of positive numbers i tending to 0. To prove Lemma
4.4 we will need the following fundamental result from [LaLi] (see Theorem 3.1,
[LaLi]).
Theorem 4.5 ([LaLi]). Let X be a Lindenstrauss space, x ∈ BX , and Y ⊂ X
be a finite-dimensional polyhedral subspace of X . Then given ε > 0 there is a
n , and
finite-dimensional subspace Z ⊂ X such that Z ⊃ Y, Z isometric to l∞
d(x, Z) < ε.
Proof of Lemma 4.4 Using Theorem 4.5 we fix a subspace Z1 ⊃ L1 isometric to
`k∞ such that d(x, Z1 ) < . Using Corollary 3.7 we fix an isometry I1 : Z1 → G
such that I1 |L1 = I. We put Z2 = I1 (Z1 ). Using Theorem 4.5 once more we fix a
subspace V2 ⊂ G isometric to `n∞1 such that Z2 ⊂ V2 and d(x, V2 ) < . We apply
Proposition 4.2(b) to spaces L2 ⊂ Z2 ⊂ V2 (note that L2 ⊂ V2 has UHB because
L2 ⊂ G has) to get subspace Z̃2 such that L2 ⊂ Z̃2 ⊂ V2 and Z̃2 ⊂ V2 has UHB.
We also get an isometry A : Z2 → Z̃2 such that A|L2 = IdL2 . Using Proposition
2.8 we find a chain of subspaces Z̃2 ⊂ D1 ⊂ D2 ⊂ · · · ⊂ Dm ⊂ V2 such that
codimD1 Z̃2 = 1, codimV2 Dm = 1 and codimDk+1 Dk = 1 for k = 1, 2, . . . , m−1
and such that each pair Dk ⊂ V2 has UHB. We inductively apply Theorem 3.1and
find an isometric embedding ψ̃ : V2 → G such that ψ̃|Z̃2 = (A ◦ I1 )−1 . We put
V1 = ψ̃(V2 ) and ψ = ψ̃ −1 .
5. D ENSITY OF SMOOTH SUBSPACES OF THE G URARIY SPACE .
The main result of this section is
Theorem 5.1. For a separable Lindenstrauss space X TFAE:
(SM) The family SF (X) of all smooth finite-dimensional subspaces of X is θ-dense
in the family F (X) of all finite-dimensional subspaces of X.
(G) The space X is the Gurariy space G.
CHARACTERISTIC PROPERTIES OF THE GURARIY SPACE
21
To prove Theorem 5.1 we need
n , n = 1, 2, ..., be a Lindenstrauss space
Lemma 5.2. Let X = cl ∪ Xn , Xn = l∞
and {ei } be from Lemma 1.4. Assume that σ ⊂ N is a finite subset of natural
numbers . Then there is an isometric embedding J : l∞ (σ) → X such that for any
i ∈ σ we have ei |l∞ (σ) = J ∗ ei .
Proof. Let p = min σ and q =Pmax σ. Let (ani )n=1,2,...,i=1,...,n be a representing
matrix of X, i.e. en+1 |Xn = ni=1 ani ei |Xn , n = 1, 2, .... Fix x = (xi )i∈σ and
define the coordinates of Jx as follows. Put
Pn1(Jx)i = 0 for i < p. Let n1 + 1 =
min{i 6∈ σ : i > p}. Define (Jx)nP
=
1 +1
i=1 an1 i xi . Next let n2 +1 = min{i 6∈
2
an2 i xi . Proceeding in this way we define
σ : i > n1 }. Define (Jx)n2 +1 = ni=1
the coordinates (Jx)i for 1 ≤ i ≤ q, i.e. the embedding of l∞ (σ) into Xq is
defined. Since Xq is embedded into X (according to the representing matrix) the
proof is complete.
Proof of Theorem 5.1. (SM)⇒(G). It follows directly from Theorem 1.7, Fact 1.8
and the following
Fact 5.3. Let M ⊂ X be a finite-dimensional smooth subspace of a Banach space
X and L ⊂ M be a proper subspace of M. Then L has property
extBX ∗ |L = BL∗
(5.53)
Proof. It is easily seen that for any f ∈ BL∗ there is h ∈ SM ∗ with h|L =
f. Since M is finite-dimensional and smooth it follows that M ∗ is strictly convex, i.e. extBM ∗ = SM ∗ . However, by the Krein-Milman theorem extBM ∗ ⊂
{extBX ∗ }|M which finishes the proof of Fact.
q
and E1 ⊃ E be a subspace of
Let E ⊂ X be a subspace of X isometric to l∞
X containing E as a hyperplane. By using (SM) find a sequence {Mn } of smooth
finite-dimensional subspaces of X which θ-converges to E1 . An easy consideration
shows that there is a subsequence {Mnk } and a sequence {Lk } of subspaces Lk ⊂
Mnk with θ − limk Lk = E. By Fact 5.3 we have
{extBX ∗ }|Lk = BLk ,
k = 1, 2, ...
(5.54)
Fix ε > 0 and let
θ(E, Lm ) < ε
(5.55)
Take f ∈ BE ∗ and let F ∈ BX ∗ be a Hahn-Banach extension of f. Put g = F |Lm
and use 5.54) to find h ∈ extBX ∗ with h|Lk = g. Next take x ∈ SE and use (5.55)
to find y ∈ SLm with ||x − y|| < ε. We have
|f (x)−h(x)| = |F (x)−h(x)| ≤ |F (x)−F (y)|+|F (y)−h(y)|+|h(y)−h(x)| ≤
||x − y|| + |g(y) − h(y)| + ||y − x|| < 2ε.
Hence ||f − h|E|| < 2ε which proves that any functional f ∈ BE ∗ may be approximated by the restrictions of the extreme points of BX ∗ . By Fact 1.8 this implies
(G) which completes the proof of (SM)⇒(G).
q
(G)⇒(SM). Let E ⊂ G be a subspace isometric to l∞
. Fix ε ∈ (0, 1), and let
∗
∗
V ⊂ E be a strictly convex symmetric body in E with
V ⊂ BE ∗ ⊂ (1 + ε)V
(5.56)
22
V.P. FONF AND P. WOJTASZCZYK
P
Next take a sequence {εp } of positive numbers with
εp < ε, and choose on
mp
the boundary ∂V a sequence of symmetric εp -nets Ap = {±hpi }i=1
such that the
following hold:
m
p+1
Ap ⊂ Ap+1 , Ap+1 \ Ap = {hp+1
}i=m
i
p +1
(5.57)
h1i = (1 − γi )ei , γi ≥ 0, i = 1, ..., q,
(5.58)
q
q
∗
where {ei }i=1 is a standard basis of l1 = E .
Next put Vp = coAp and denote Ep a Banach space E with norm ||x||p =
max x(Vp ). Choosing sets Ap in an appropriate way we can get the additional
property:
Vp ⊂ Vp+1 ⊂ (1 + 2εp )Vp
(5.59)
Note that from (5.56) it easily follows that
V1 ⊂ BE ∗ ⊂ (1 + 3ε)V1
(5.60)
m
Now we define a sequence of embeddings of spaces E and Ep into spaces l∞p
m
m
and a sequence of embeddings of l∞p into l∞p+1 , p = 1, 2, ...
m1 as follows
First define φ : E → l∞
(φ(x))j = (1 − γj )−1 h1j (x),
j = 1, ..., q, x ∈ E,
(φ(x))j = h1j (x), j = q + 1, ..., x ∈ E.
m1 as follows
Next define ψ1 : E1 → l∞
(ψ1 (x))j = h1j (x),
j = 1, ..., m1 , x ∈ E1 .
Clearly, both φ and ψ1 are isometric embeddings and from (5.60) it easily follows
that θ(φ(E), ψ1 (E1 )) ≤ 3ε.
m1 → lm2 and an isometric
Next we define an isometric embedding ξ1 : l∞
∞
m
2
embedding ψ2 : E2 → l∞ . From (5.59) follows that ||h2i ||1 ≤ 1 + 2ε1 , i =
m1 + 1, ..., m2 . Hence
h2i
=
m1
X
a1j h1j ,
j=1
p1
X
|a1j | ≤ 1 + 2ε1 , i = m1 + 1, ..., m2 .
j=1
Define
(ξ1 (x))i = (x)i , i = 1, ..., m1 ,
m1
X
−1
m1
(ξ1 (x))i = (1 + ε1 )
a1j (x)j , i = m1 + 1, ..., m2 , x ∈ l∞
,
j=1
(ψ2 (x))j = h2j (x),
j = 1, ..., m2 , x ∈ E2 .
Clearly, the compositions ξ1 φ and ξ1 ψ1 are isometric embeddings and
θ(ξ1 φ(E), ξ1 ψ1 (E1 )) ≤ 2ε,
θ(ξ1 ψ1 (E1 ), ψ2 (E2 )) ≤ 2ε1 .
m2 → lm3 and an isometric emNext we define an isometric embedding ξ2 : l∞
∞
m
bedding ψ3 : E3 → l∞3 . From (5.59) follows that ||h3i ||2 ≤ 1 + 2ε2 , i =
m2 + 1, ..., m3 . Hence
h3i =
m2
X
j=1
a2j h2j ,
p2
X
j=1
|a2j | ≤ 1 + 2ε2 , i = m2 + 1, ..., m3 .
CHARACTERISTIC PROPERTIES OF THE GURARIY SPACE
23
Define
(ξ2 (x))i = (x)i , i = 1, ..., m2 ,
m2
X
−1
m2
(ξ2 (x))i = (1 + ε2 )
a2j (x)j , j = m2 + 1, ..., m3 , x ∈ l∞
,
j=1
(ψ3 (x))j = h3j (x),
j = 1, ..., m3 , x ∈ E3 .
Clearly,
θ(ξ2 ξ1 φ(E), ξ2 ξ1 ψ1 (E1 )) ≤ 2ε,
θ(ξ2 ξ1 ψ1 (E1 ), ξ2 ψ2 (E2 )) ≤ 2ε1 ,
θ(ξ2 ψ2 (E2 ), ψ3 (E3 )) ≤ 2ε2 .
mp
The further construction is clear. Let Y be the completion of ∪∞
p=1 l∞ and τp :
mp
mp
l∞ → Y, p = 1, 2, ..., be natural embeddings of l∞ into Y. If
Ẽ = τ1 φ(E), Lp = τp ψp (Ep ), p = 1, 2, ...,
then
θ(Ẽ, L1 ) ≤ 2ε, θ(Lp , Lp+1 ) ≤ 2εp , p = 1, 2, ....
Therefore the sequence of subspaces {Lp } θ-converges to, say L. It is easily seen
that θ(Ẽ, L) ≤ 6ε, and that L is smooth. Indeed, it is well-known that if θ −
limp Lp = L then the sequence Lp tends to L in the Banach-Mazur metric too.
However, the sequence Lp tends in Banach-Mazur metric to the space E with the
norm |||x||| = max x(V ), where V is strictly convex. Hence, L is smooth.
Finally we should embed Y into G. We want to use Theorem 1.6. The easiest
way to proceed is the following. Apply the above procedure of constructing Y just
using, say even “coordinates”, and then “fill” the odd coordinates by using a matrix
A as above. The proof is complete.
R EFERENCES
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