Rev. R. Acad. Cien. Serie A. Mat.
VOL.100 (1-2), 2006, pp. 295–323 An´alisis Matem´atico / Mathematical Analysis Art´ıculo panor´amico / Survey
Embedding into Banach spaces with finite dimensional decompositions
Edward W. Odell and Thomas Schlumprecht
Abstract. This paper deals with the following types of problems: Assume a Banach spaceXhas some property (P). Can it be embedded into some Banach spaceZ with a finite dimensional decomposition having property (P), or more generally, having a property related to (P)? Secondly, given a class of Banach spaces, does there exist a Banach space in this class, or in a closely related one, which is universal for this class?
Inclusi ´on en espacios de Banach con descomposiciones finito-dimensionales
Resumen. Este art´ıculo trata los siguientes tipos de problemas: se supone que un espacio de BanachX tiene cierta propiedad (P), ¿puede incluirse en un espacio de BanachZcon una descomposici´on finito- dimensional que verifique (P), o, en general, una propiedad relacionada con (P)? En segundo lugar, dada una clase de espacios de Banach, ¿existe en dicha clase, o en una pr´oxima, un espacio de Banach universal para la clase?
1. Introduction
The fact that every separable infinite dimensional real Banach spaceX embeds intoC[0,1]dates back to the early days of Banach space theory [3, Th´eor`eme 9, page 185]. This result has inspired two types of problems. First, given a spaceXin a certain class can it be embedded isomorphically into a spaceY of the same class with a basis or, more generally, a finite dimensional decomposition (FDD)? Secondly, given a class of spaces does there exist a universal spaceXfor that class which is in the class or in a closely related one? By sayingXis universal for a classCwe mean that eachY ∈Cembeds intoX. As it happens these two types of problems are often related in that solving a problem of the first type can lead to a solution to the analogous problem of second type.
For example, J. Bourgain [4] asked if there exists a separable reflexive spaceXwhich is universal for the class of all separable superreflexive Banach spaces. This question arose from his result that ifX contains an isomorph of all separable reflexive spaces thenXis universal, i.e., contains an isomorph ofC[0,1]. This improved an earlier result of Szlenk [29] who showedX∗was not separable. Work by S. Pruss [28] showed that it sufficed to prove that for a separable superreflexive spaceY there exists1< q ≤p < ∞,C <∞
Presentado por Vicente Montesinos Santaluc´ıa.
Recibido: 09/03/2006.Aceptado: 15/03/2006.
Palabras clave / Keywords: Finite-dimensional decompositions; universal spaces; isomorphical embeddings; Szlenk index.
Mathematics Subject Classifications: 46B03; 46B20.
°c 2006 Real Academia de Ciencias, Espa˜na.
and a spaceZwith an FDDE= (Ei)satisfyingC-(p, q)-estimates, C−1µ X
kzikp
¶1/p
≤ kX
zik ≤Cµ X kzikq
¶1/q
for all block sequences(zi)ofZw.r.t.(Ei). Such a spaceZis automatically reflexive and thus we have the problem of givenp, q, when does a reflexive spaceY embed into such a spaceZ.
An earlier version of this problem (whenp = q) was raised by W.B. Johnson [9] resulting from his work onLpand earlier work with M. Zippin [14, 15]. The problem addressed in [9] was to characterize when a subspaceX of Lp,1 < p < 2, embeds into`p. In [12] it was shown that if a subspaceX of Lp, with2 < p < ∞, embeds into `p if (and only if by [16])X does not contain an isomorph of `2
(later improved toXalmost isometrically embeds into`p[18]). This characterization does not work inLp, 1 ≤ p < 2, sinceLq embeds intoLp ifp ≤q ≤2, but thep > 2characterization is equivalent (again by [16]) to every normalized basic sequence inX has a subsequence 2-equivalent to the unit vector basis of`p. Johnson showed that this criterion (with “2-equivalent” replaced byC-equivalent for someC <∞) characterized whenX ⊆Lp,1 < p < 2, embeds into`p. His argument showed thatX embedded into (P
Hn)`pfor some blocking of the Haar basis into anFDD(Hn)and of course(P
Hn)`pembeds into`p. Johnson also considered the dual problem which brought quotient characterizations into the picture. These had appeared earlier [15] when it was shown thatX embeds into(P
En)`p, where(En)is a sequence of finite dimensional Banach spaces iffXis a quotient of such a space.
It turns out that the characterization required to ensure that a reflexive spaceX embeds into one with an FDD satisfying(p, q)-estimates is not a subsequence criterion in the general setting, i.e. if we do not assumeX to be a subspace ofLp, but rather one that can be expressed in terms of weakly null trees inSX, the unit sphere ofX. This can be viewed as an infinite version of the notion of asymptotic structure [23].
IfX is a Banach space then, forn ∈ N, a normalized monotone basis is said to be in thenth-asymptotic structure ofX, and we write(ei)ni=1∈ {X}n, if for allε >0the following holds (cof(X)will denote the set of all closed subspaces ofX having finite codimension):
∀X1∈cof(X)∃x1∈SX1∀X2∈cof(X)∃x2∈SX2 . . .∀Xn∈cof(X)∃xn∈SXn (1..1) (xi)ni=1is(1 +ε)-equivalent to(ei)ni=1.
The fact that some normalized monotone basis(ei)ni=1is a member of{X}ncan be, maybe more intuitively, described by a game between two players. Player I choosesX1∈cof(X), then Player II choosesx1∈SX1. This procedure is repeated until a sequence(xi)ni=1is obtained. Player II is declared winner of the game if (xi)ni=1is(1 +ε)-equivalent to(ei)ni=1. Condition (1..1) means that Player II has a winning strategy.
It is not hard to show that{X}n is a compact subset ofMn, the set of all such normalized monotone bases(ei)ni=1under the metriclogdb(·,·)wheredb((ei)ni=1,(fi)ni=1)is the basis equivalence constant be- tween the bases. Lembergs [20] proof of Krivine’s theorem shows that there is a1 ≤p≤ ∞, so that the unit vector basis of`np is in{X}nfor alln∈N. In [23] it is shown that{X}nis also the smallest closed subsetCofMnwith the property that, for allε >0, player I has a winning strategy for forcing player II to select(xi)ni=1withdb((xi)ni=1,C)<1 +ε. This does not generalize to produce say{X}∞since we lose compactness. However we can still consider a classAof normalized monotone bases with the property that in the infinite game player I has a winning strategy for forcing II to select(xi)∞i=1∈ A.
These notions can be restated in terms of weakly null trees when X∗ is separable. Indeed {X}n
is the smallest class such that every weakly null tree of length nin SX admits a branch (xi)ni=1 with db((xi)ni=1,{X}n) < 1 +ε. Precise definitions of weakly null trees and other terminology appear in Section 2.
IfAis as above forXwe can also restate the winning strategy for player I in terms of weakly null trees (of infinite level) but there are some difficulties. First given playsX1, X2, . . .by player I we cannot select a branch(xi)withxi∈Xifor allibut only thatxiis close to an element ofSXi. Secondly not all games are determined so we need a fatteningAεofAand then need to close it toAεin the product of the discrete
topology onSX to obtain a determined game. This will lead to the property that if every weakly null tree inX admits a branch inAthen ifX ⊆Z, a space with an appropriate FDD(Ei), one can find a blocking (Fi)of(Ei)and¯δ= (δ),δi ↓0, so that every(xi)⊆SX which is aδ-skipped block sequence w.r.t.¯ (Fi) is inAε. These will be defined precisely in Section 2.
An application will be the solution of Johnson’s problem (when does a reflexive spaceX embed into an`p-FDD?)), Pruss’ problem (when does a reflexive spaceX embed into one with an FDD satisfying (p, q)-estimates) and, as a consequence, Bourgain’s problem. These solutions will be given in Sections 4 and 5. Among other characterizations we will show that if for someC < ∞every weakly null tree in a reflexive spaceXadmits a branchC-dominating the unit vector basis of`pand a branchC-dominated by the unit vector basis of`qthenX embeds into a space with an FDD satisfying(p, q)-estimates.
The machinery developed in Section 2 also has applications in the nonreflexive setting. In Section 3 we consider and characterize spacesX of Szlenk indexω, the smallest possible. Sz(X)is an ordinal index which is less thanω1iffX∗is separable. Forε >0setK0(X, ε) =BX∗and forα < ω1we recursively define
Kα+1(X, ε) =
½
x∗∈Kα(X, ε) : ∃(x∗n)⊆Kα(x, ε)with
w∗−limn→∞x∗n =x∗and lim infn→∞kx∗n−x∗k ≥ε
¾
Ifαis a limit ordinal,
Kα(X, ε) = \
β<α
Kβ(X, ε). SZ(X, ε)is the smallestαwithKα(X, ε) =∅orω1otherwise.
SZ(X) = sup{SZ(X, ε) : 0< ε <1}.
We will show thatSz(X) = ω iffX∗ can be embedded as aw∗-closed subspace of a spaceZ with an FDD satisfying1-(p,1)-estimates. A long list of further equivalent conditions (Theorem 3.4) will be given including that X can be renormed to be w∗-uniform Kadec Klee and X can be renormed to be asymptotically uniformly smooth (of power typeqfor someq >1).
Asymptotic uniformly smooth (a.u.s.) and asymptotic uniformly convex (a.u.c) norms, defined in Sec- tion 3, are asymptotic versions of uniformly smooth and uniformly convex due to [11] based upon modulii of V.D. Milman [22]. Theorem 3.4, mentioned above, gives the result thatX can be given an a.u.s. norm iff it can be given one of power typeqfor someq > 1. We obtain a similar result for a.u.c. for reflexive spaces. Recall that Pisier [27] proved that a superreflexive (equivalently, uniformly convex) space can be renormed to be uniformly convex of power typepfor some2≤p <∞and similarly for uniformly smooth with1< p≤2.
In Section 3 we also give a proof of Kalton’s theorem [17] that a Banach spaceX embeds intoc0if for someC < ∞every weakly null tree inSXadmits a branch(xi)∞i=1satisfyingsupnkPn
1xik ≤C. This proof fits nicely into our Section 2 machinery.
In Section 5 we discuss applications of our results to universal problems. In regard to Bourgain’s problem we show the space constructed is universal for the class
{X :Xis reflexive,Sz(X) =Sz(X∗) =ω},
which includes all superreflexive spaces. We also discuss the universal problem for reflexive a.u.s. (or a.u.c.) spaces.
A central theme of the problems we have presented is coordinatization. A coordinate-free property is considered and we wish to embed a spaceXwith this property into a spaceZwith an FDD which realizes this property w.r.t. its “coordinates”. The tools we use, in addition to the ones mentioned above, are several.
There are the blocking arguments of Johnson and Zippin [9], [14, 15] and some known embedding theorems which we cite now.
1.1 [5]
IfX∗is separable thenXis a quotient of a space with a shrinking basis.
1.2 [30]
IfX∗is separable thenXembeds into a space with a shrinking basis.
1.3 [30]
IfX is reflexive thenXembeds into a reflexive space with a basis.
We will often begin withX ⊆Z, one of the spaces given by 1.2, 1.3 or withX a quotient ofZ (as in 1.1) and the problem will be to put a new norm onZwhich reflects the structure ofX that we wish to coordinatize and maintains thatXis a subspace ofZ(or a quotient).
All of our Banach spaces in this paper are real and separable. We will use X, Y, Z, . . .for infinite dimensional spaces andE, F, G, . . .for finite dimensional spaces.
Most of the results we will present have appeared in a number of recent papers ([24], [25], [26] [19], [17], [7], [11]). As the theory has developed the proofs and results have been better understood, generalized and improved. Our aim is to give a unified presentation of these improvements and in several cases present easier proofs. New results are also included.
2. A general combinatorial result
In this section we state and prove three general combinatorial results (Theorem 1 and Corollaries 1 and 2). There statement are reformulations and improvements of results in [24]. We will present a different, possibly more accessible, proof.
We first introduce some notation.
LetZbe a Banach space with an FDDE= (En). Forn∈Nwe denote then-thcoordinate projection byPnE, i.e. PnE : Z →En, P
zi 7→ zn.For finiteA ⊂Nwe putPAE = P
n∈APnE. Theprojection constant of(En)(inZ) is defined by
K=K(E, Z) = sup
m≤nkP[m,n]E k.
Recall thatK is always finite and, as in the case of bases, we call(En)bimonotone (inZ)ifK = 1.
By passing to the equivalent norm
||| · |||:Z →R, z7→ sup
m≤nkP[m,n]E (z)k, we can always renormZ, so thatK= 1.
For a sequence(Ei)of finite dimensional spaces we define the vector space c00(⊕∞i=1Ei) =©
(zi) :zi∈Ei, fori∈N,and{i∈N:zi6= 0}is finiteª ,
which is dense in each Banach space for which(En)is an FDD. ForA⊂Nwe denote by⊕i∈AEithe linear subspace ofc00(⊕Ei)generated by the elements of(Ei)i∈Aand we denote its closure inZ by(⊕Ei)Z. As usual we denote the vector space of sequences inRwhich are eventually zero byc00and its unit vector basis by(ei).
The vector spacec00(⊕∞i=1Ei∗), whereEi∗is the dual space ofEi, fori∈N, is aw∗-dense subspace of Z∗. We denote the norm closure ofc00(⊕∞i=1Ei∗)inZ∗byZ(∗).Z(∗)isw∗-dense inZ∗, the unit ballBZ(∗)
normsZand(Ei∗)is an FDD ofZ(∗)having a projection constant not exceedingK(E, Z). IfK(E, Z) = 1 thenBZ(∗)is 1-norming andZ(∗)(∗)=Z.
Forz∈c00(⊕Ei)we define theE-support ofzby suppE(z) =©
i∈N:PiE(z)6= 0ª .
A non-zero sequence (finite or infinite)(zj)⊂c00(⊕Ei)is called ablock sequence of(Ei)if max suppE(zn)<min suppE(zn+1), whenevern∈N(orn <length(zj)), and it is called askipped block sequence of(Ei)if
1<min suppE(z1)<max suppE(zn)<min suppE(zn+1)−1, whenevern∈N(orn <length(zi)).
Letδ= (δn)⊂(0,1]. A (finite or infinite) sequence(zj)⊂SZ ={z ∈Z :kzk= 1}is called aδ-block sequence of(En)or aδ-skipped block sequence of(En)if there are1≤k1< `1< k2< `2< . . .inNso that
kzn−P[kEn,`n](zn)k< δn, orkzn−P(kEn,`n](zn)k< δn, respectively,
for alln ∈ N(orn ≤length(zj)). Of course one could generalize the notion ofδ- block andδ-skipped block sequences to more general sequences, but we prefer to introduce this notion only for normalized sequences. It is important to note that in the definition ofδ-skipped block sequencesk1 ≥ 1, and that therefore theE1-coordinate ofz1is small (depending onδ1).
A sequence of finite-dimensional spaces(Gn)is called ablocking of(En)if there are0 =k0< k1 <
k2< . . .inNso thatGn=⊕ki=kn n−1+1Ei, forn= 1,2. . ..
We denote the sequences inSZ of length n∈ NbySZn and the infinite sequences inSZ bySZω. For m, n∈N, forx= (x1, x2, . . . xm)∈SZmandy = (y1, y2, . . . yn)∈SZnory= (yi)∈SZωwe denote the concatenation ofxandyby(x, y), i.e.
(x, y) = (x1, x2. . . xm, y1, . . . ym), or(x, y) = (x1, x2. . . xm, y1, y2. . .)respectively. We also allow the casex=∅ory=∅and let(∅, y) =yand(x,∅) =x.
LetA ⊂SZωbe given. We denote the closure ofAwith respect to the product topology of the discrete topology onSZ byA. Note that ifAis closed it follows forx= (xi)∈SZw,
x∈ A ⇐⇒ ∀n∈N∃z∈SωZ (x1, x2, . . . xn, z)∈ A (2..1) Ifε= (εi)is a sequence in[0,∞)we write
Aε=©
(zi)∈SZω:∃(˜zi)∈ A, kzi−z˜ik ≤εi
ª
and call the setAεtheε-fattening ofA. For` ∈ Nandε = (εi)`i=1 ⊂ [0,∞)we letAε = Aδ, where δ= (δi)andδi=εi, fori= 1,2. . . `andδi= 0ifi > `.
If`∈Nandx1, x2, . . . x`∈SZ we let A(x1, x2, . . . x`) =©
z= (zi)∈SZω: (x1, x2, . . . x`, z)∈ Aª . LetA ⊂SZωandB=Q∞
i=1Bi, whereBn⊂SZforn∈N.
We consider the following(A,B)-game between two players: Assume thatE= (Ei)is an FDD forZ. Player I choosesn1∈N
Player II choosesz1∈c00
¡⊕∞i=n1+1Ei
¢∩B1, Player I choosesn2∈N
Player II choosesz2∈c00
¡⊕∞i=n2+1Ei
¢∩B2, ...
Player I wins the(A,B)-game if the resulting sequence(zn)lies inA. If Player I has a winning strategy (forcing the sequence(zi)to be inA) we will writeW I(A,B)and if Player II has a winning strategy (being able to choose(zi)outside ofA) we writeW II(A,B). IfAis a Borel set with respect to the product of the discrete topology onSZω(note thatBis always closed in the product of the discrete topology onSZω), it follows from the main theorem in [21] that the game is determined, i.e. eitherW I(A,B)orW II(A,B).
Let us defineW II(A,B)formally. We will need to introduce trees in Banach spaces.
We define
T∞= [
`∈N
©(n1, n2, . . . , n`) :n1< n2< . . . n`are inNª .
Ifα= (m1, m2, . . . m`)∈T∞, we call`thelength ofαand denote it by|α|, andβ = (n1, n2, . . . nk)∈ T∞is called anextension ofα, orαis calleda restriction of β, ifk≥`andni=mi, fori= 1,2, . . . , `.
We then writeα≤βand with this order(T∞,≤)is a tree.
In this work trees in a Banach space X are families inX indexed by T∞, thus they are countable infinitely branching trees of countably infinite length.
For a tree(xα)α∈T∞in a Banach spaceX, andα= (n1, n2, . . . , n`)∈T∞∪ {∅}we call the sequences of the form(x(α,n))n>n` nodes of(xα)α∈T∞. The sequences(yn), with yi = x(n1,n2,...,ni), fori ∈ N, for some strictly increasing sequence(ni) ⊂N, are calledbranches of (xα)α∈T∞. Thus, branches of a tree(xα)α∈T∞ are sequences of the form(xαn)where(αn)is a maximal linearly ordered (with respect to extension) subset ofT∞.
If(xα)α∈T∞ is a tree in X and ifT0 ⊂ T∞is closed under taking restrictions so that for eachα ∈ T0 ∪ {∅} infinitely many direct successors of αare also in T0 then we call(xα)α∈T0 afull subtree of (xα)α∈T∞. Note that (xα)α∈T0 could then be relabelled to a family indexed by T∞ and note that the branches of (xα)α∈T0 are branches of (xα)α∈T∞ and that the nodes of (xα)α∈T0 are subsequences of certain nodes of(xα)α∈T∞.
We call a tree(xα)α∈T∞in a Banach spaceXnormalizedifkxαk= 1, for allα∈T∞andweakly null if every node is weakly null. More generally ifT is a topology onX and a tree(xα)α∈T∞ in a Banach spaceXis calledT-null if every node converges to0with respect toT.
If(xα)α∈T∞ is a tree in a Banach spaceZwhich has an FDD(En)we call it ablock tree of (En)if every node is a block sequence of(En).
We will also need to consider trees of finite length. For`∈Nwe call a family(xα)α∈T∞,|α|≤`inX a tree of length`. Note that the notions nodes, branches,T-null and block trees can be defined analogously for trees of finite length.
Definition 1. Assume thatZ is a Banach space with an FDD(Ei),A ⊂ SωZ andB = Q∞
i=1Bi, with Bi⊂SZfori∈N. We say thatPlayer II has a winning strategy for the(A,B)-gameif
(W II(A,B)) There exists a block tree(xα)α∈T∞of(Ei)inSZ all of whose branches are inBbut none of its branches are inA.
In case that the(A,B)-game is determinedW I(A,B)can be therefore stated as follows.
(W I(A,B)) Every block tree(xα)α∈T∞of(Ei)inSXall of whose branches are inBhas a branch inA.
The proof of the following Proposition is easy.
Proposition 1. LetA,A ⊂e SZω,B =Q∞
i=1Bi, withBi ⊂SZ fori ∈N. Assume that the(A,B)-game and the(A,e B)-game are determined.
a) IfA ⊂A, thene
W I(A,B)⇒W I(A,eB)andW II(A,eB)⇒W II(A,B).
b) W I(A,B) ⇐⇒ ∃n∈N∀x∈¡
⊕∞i=n+1Ei
¢∩B1 W I((A(x),Q∞
i=2Bi)
c) If`∈N,ε= (εi)`i=1⊂[0,∞)andxi, yi∈Biwithkxi−yik ≤εifori= 1,2. . . `then W I³
A(x1, x2, . . . , x`), Y∞ i=`+1
Bi
´
⇒W I³
Aε(y1, y2, . . . , y`), Y∞ i=`+1
Bi
´ .
Lemma 1. LetA, andε= (εi), δ= (δi)⊂[0,∞)Then
¡Aε
¢
δ⊂ Aε+δ. PROOF. We observe
u= (ui)∈¡ Aε
¢
δ
⇐⇒ ∀n∈N∃v(n)∈SZω (u1, . . . , un, v(n))∈¡ Aε
¢
δ
⇒∀n∈N∃x1, x2, . . . xn∈SZandw(n)∈SωZ
kxi−uik ≤δi, fori= 1, . . . n, and(x1, . . . , xn, w(n))∈ Aε,
⇒∀n∈N∃x1, x2, . . . xn∈SZandw(n)∈SωZ∀m∈N∃y(m)∈SωZ
kxi−uik ≤δifori= 1,2. . . nand(x1, . . . , xn, w(n)1 , w2(n). . . wm(n), y(m))∈ Aε
=⇒∀n∈N∃x1, x2, . . . xn∈SZ∃y(n)∈SZω
kxi−uik ≤δifori= 1,2. . . nand(x1, . . . , xn, y(n))∈ Aε
=⇒∀`∈N∃z(`)∈ A kui−zi(`)k ≤δi+εi, fori= 1,2. . . `
⇐⇒u∈ Aε+δ.
Now we can state one of our main combinatorial principles.
Theorem 1. Let Z have an FDD(Ei)and letBi ⊂ SZ, fori = 1,2. . .. PutB = Q∞
i=1Bi and let A ⊂SZω.
Assume that for allε= (εi)⊂(0,1]we haveW I(Aε,B).
Then for allε= (εi)⊂(0,1]there exists a blocking(Gi)of(Ei)so that every skipped block sequence (zi)of(Gi), withzi∈Bi, fori∈N, is inAε.
PROOF. Letε= (εi)⊂(0,1]be given. Fork= 0,1,2. . .putε(k)= (ε(k)i )with ε(k)i =εi(1−2−k)/2fori∈N.
We putAe=Aε/2.
For`∈Nwe writeB(`)=Q∞
i=`+1Bi.
By induction we choose fork∈Nnumbersnk ∈Nso that0 =n0 < n1 < n2 < . . ., and so that for anyk∈N, putting withGk=⊕ni=nk k−1Ei,
W I¡Aeε(k)(σ, x),B(`+1)¢
for any0≤` < kand any normalized skipped block (2..2) σ= (x1, x2, . . . x`)∈
Y` i=1
Biof(Gi)k−1i=1 (σ=∅if`= 0) andx∈S⊕∞i=nk+1Ei∩B`+1
W I¡
Aeε(k)(σ),B(`)¢
for any0≤` < kand any normalized skipped block (2..3) σ= (x1, x2, . . . x`)∈
Y` i=1
Biof(Gi)ki=1
Fork= 1we deduce from Proposition 1 (b) that there is ann1 ∈Nso thatW I¡Aeε(1)(x),B(1)¢ for any x∈S⊕∞i=n
1+1Ei ∩Bi. This implies (2..2) and (2..3) (note that fork = 1σcan only be chosen to be∅in (2..2) and (2..3)).
Assume n1 < n2 < . . . nk have been chosen for somek ∈ N. We will first choose nk+1 so that (2..2) is satisfied. In the case thatk = 1we simply choosen2 = n1+ 1and note that (2..2) fork = 2 follows from (2..2) fork = 1since in both cases σ = ∅ is the only choice. Ifk > 1 we can use the compactness of the sphere of a finite dimensional space and choose a finite setF of normalized skipped blocks(x1, x2, . . . , x`)∈ Q`
i=1Bi, of(Gi)ki=1so that for any` ≤ kand any normalized skipped block with length`,σ = (x1, x2, . . . , x`) ∈ Q`
i=1Bi of(Gi)ki=1, there is aσ0 = (x01, x02, . . . , x0`) ∈ F with kxi−x0ik< εi2−k−2, fori= 1,2, . . . , `. Then, using the induction hypothesis (2..3) fork, and Proposition 1 (b), we choose nk+1 ∈ Nlarge enough so that W I¡
Aeε(k)(σ, x),B(`+1)¢
for any σ ∈ F and x ∈ S⊕∞i=nk+1+1Ei∩B`+1. From Proposition 1 (c) and our choice ofFwe deduceW I¡Aeε(k+1)(σ, x),B(`+1)¢ for any0 ≤ ` < k, any normalized skipped block σof (Gi)ki=1 of length `inQ`
i=1Bi and any x ∈ S⊕∞i=nk+1+1Ei∩B`+1, and, thus, (using the induction hypothesis forσ=∅) we deduce (2..2) fork+ 1.
In order to verify (2..3) letσ= (x1, x2, . . . , x`)∈Q`
i=1Bibe a normalized skipped block of(Gi)k+1i=1 (the case σ = ∅ follows from the induction hypothesis). Then σ0 = (x1, x2, . . . , x`−1)is empty or a normalized skipped block sequence of(Gi)k−1i=1 inQ`−1
i=1Bi. In the second caseW I¡Aeε(k+1))(σ),B(`)¢
= W I¡
Aeε(k+1))(σ0, x`),B(`)¢
follows from (2..2) forkand from Proposition 1 (a). This finishes the recursive definition of thenk’s andGk’s.
Let(zn)any normalized skipped block sequence of(Gi)which lies inB. For anyn∈Nit follows from (2..3) forσ= (zi)ni=1thatW I(Aeε/2(σ),B), and, thus,Aeε/2(σ)6=∅, which means thatσis extendable to a sequence inAeε/2(note thatlimn→∞ε(n)i =εi). Thus, any normalized skipped block sequence which is element ofBlies inAeε/2and, thus, by Lemma 1, inAε.
Now letX be a closed subspace ofZ having an FDD(Ei)andA ⊂ SXω. We consider the following game
Player I choosesn1∈N Player II choosesx1∈¡
⊕∞i=n1+1Ei
¢
Z∩X, kx1k= 1, Player I choosesn2∈N
Player II choosesx2∈¡
⊕∞i=n2+1Ei¢
Z∩X, kx2k= 1, ...
As before, Player I has won if(xi)∈ A. Since the game does not only depend onAbut on the superspace Zin whichX is embedded and its FDD(EI)we denote the game(A, Z)-game.
Definition 2. Assume thatX is the subspace of a spaceZwhich has an FDD(Ei)and thatA ⊂ SXω. Define forn∈N
Xn=X∩¡
⊕∞i=n+1Ei
¢
Z ={x∈X:∀z∗ ∈ ⊕ni=1Ei∗ z∗(x) = 0}¢ , a closed subspace with finite codimension inX.
We say thatPlayer II has a winning strategy in the(A, Z)-game if
W II(A, Z) there is a tree (xα)α∈T∞ in SX so that for any α = (n1, . . . n`) ∈ T∞ ∪ ∅ x(α,n) ∈ Xnwhenevern > n`, and so that no branch lies inA.
In case that the (A, Z)-game is determined Player I has a winning strategy in the (A, Z)-game if the negation ofW II(A, Z)is true and thus
W I(A, Z)For any tree(xα)α∈T∞inSXso that for anyα= (n1, . . . n`)∈T∞∪ ∅x(α,n)∈Xnwhenever n > n`, there is branch inA.
ForA ⊂SωX ⊂SZωand a sequenceε= (εi)in[0,∞)we understand byAεtheε-fattening ofAas a subset ofSZω. In case we want to restrict ourselves toSXwe writeAXε , i.e.
AXε =Aε∩SXω =©
(xi)∈SωX:∃(yi)∈ A kxi−yik ≤εifor alli∈Nª .
SinceSXω is closed inSZωwith respect to the product of the discrete topology, we deduce thatAX =AX forA ⊂SXω.
The following Proposition reduces the(A, Z)-game to a game we treated before. In order to be able to do so we need some technical assumption on the embedding ofX intoZ(see condition (2..4) below).
Proposition 2. LetX⊂Z, a space with an FDD(Ei). Assume the following condition onX,Zand the embedding ofX intoZis satisfied:
There is aC >0so that for allm∈Nandε >0there is ann=n(ε, m)≥m (2..4) kxkX/Xm ≤C£
kP[1,n]E (x)k+ε¤
wheneverx∈SX.
Assume thatA ⊂SXω and that for all null sequencesε⊂(0,1]we haveW I(AXε , Z).
Then it follows for all null sequencesε= (εi)⊂(0,1]thatW I(Aε,(SXω)δ)holds, whereδ= (δi)with δi=εi/28CKfori∈N, withCsatisfying (2..4) andKbeing the projection constant of(Ei)inZ.
PROOF. LetA ⊂ SXω and assume thatW I(AXη, Z)is satisfied for all null sequencesη = (ηi) ⊂ (0,1]. For a null sequenceε= (εi)⊂(0,1]we need to verifyW I(Aε,(SXω)δ)(withδi =εi/28KC for i∈N) and so we let(zα)α∈T∞ be a block tree of(Ei)inSZall of whose branches lie in(SXω)δ ={(zi)∈ SZω: dist(zi, SX)≤δifori= 1,2. . .}.
After passing to a full subtree of(zα)we can assume that for anyα= (m1, . . . m`)inT∞
zα∈ ⊕∞j=1+n(δ`,m`)Ej (2..5)
(wheren(ε, m)is chosen as in (2..4)).
Forα= (m1, m2, . . . m`)∈T∞we chooseyα∈SXwithkyα−zαk<2δ`and, thus, by (2..5) kP[1,n(δE `,m`)](yα)k=kP[1,n(δE `,m`)](yα−zα)k ≤2Kδ`.
Using (2..4) we can therefore choose anx0α∈Xm` so that
kx0α−yαk ≤C(2Kδ`+δ`)≤3CKδ`, and thus
1−3CKδ`≤ kx0αk ≤1 + 3CKδ`. Lettingxα=x0α/kx0αkwe deduce that
kyα−xαk ≤ kyα−x0αk+kx0α−xαk
≤3CKδ`+ (1 + 3CKδ`)3CKδ`/(1−3CKδ`)≤12CKδ`
(the last inequality follows from the fact that(1 + 3CKδ`)/(1−3CKδ`)≤3) and, thus, kzα−xαk ≤14CKδ`=ε`/2.
UsingW I(AXε/2, Z)and noting thatxα∈Xm`, forα= (m1, m2, . . . m`)∈T∞we can choose a branch of(xα)which is inAXε/2. Thus, the corresponding branch of(zα)lies inAε.
From [24, Lemma 3.1] it follows that every separable Banach spaceX is a subspace of a spaceZwith an FDD satisfying the condition (2..4) (withn(m) =m). The following Proposition exhibits two general situations in which (2..4) is automatically satisfied.
Proposition 3. AssumeX is a subspace of a spaceZhaving an FDD(Ei). In the following two cases (2..4) holds:
a) If(Ei)is a shrinking FDD forZ. In that caseCin (2..4) can be chosen arbitrarily close to1.
b) If(Ei)is boundedly complete forZ (i.e. Z is the dual ofZ(∗)) and the ball ofX is a w∗-closed subset ofZ. In that caseCcan be chosen to be the projection constantKof(Ei)inZ.
PROOF. In order to prove (a) we will show that for anym ∈ Nand any0 < ε < 1 there is an n=n(ε, m)so that
kxkX/Xm ≤(1 +ε)£
kP[1,n]E (x)k+ε¤
, wheneverx∈SX
(i.e.Cin (2..4) can be chosen arbitrarily close to1).
SinceX/Xmis finite dimensional and
(X/Xm)∗=Xm⊥ =©
x∗∈X∗ :x∗|Xm ≡0ª , we can choose a finite setAm⊂SXm⊥ ⊂SX∗for which
kxkX/Xm ≤(1 +ε) max
f∈Am
|f(x)|wheneverx∈X.
By the Theorem of Hahn Banach we can extend eachf ∈Amto an elementg ∈SZ∗. Let us denote the set of all of these extensionsBm. SinceBmis finite and since(Ei∗)is an FDD ofZ∗ we can choose an n = n(ε, m)so thatkP[1,n(m)]E∗ (g)−gk < εfor allg ∈ Bm. SinceP[1,n(m)]E∗ is the adjoint operator of P[1,n(m)]E (considerP[1,n(m)]E∗ to be an operator fromZ∗toZ∗andP[1,n(m)]E to be an operator fromZtoZ), it follows forx∈SX, that
kxkX/Xm ≤(1 +ε) max
g∈Bm
|g(x)|
≤(1 +ε) max
g∈Bm
£¯¯P[1,n(m)]E∗ (g)(x)¯
¯+kP[1,n(m)]E∗ (g)−gk¤
≤(1 +ε)£
g∈Bmaxm
|g¡
P[1,n(m)]E (x)¢
|+ε¤
≤(1 +ε)£
kP[1,n(m)]E (x)k+ε¤ , which proves our claim and finishes the proof of part (a).
In order to show (b) we assume thatXis a subspace of a spaceZwhich has a boundedly complete FDD (Ei)and the unit ball ofXis a w∗-closed subset ofZ, which is the dual ofZ(∗).
Form∈Nandε >0we will show that the inequality in (2..4) holds for somenandC=K. Assume that this was not true. and we could choose a sequence(xn)⊂SXso that for anyn∈N
kxnkX/Xm > K£
kP[1,n]E (xn)k+ε¤ .
By the compactness ofBX in the w∗ topology we can choose a subsequencexnk which converges in w∗ to somex∈BX. For fixed`it follows that(P[1,`]E (xnk))converges in norm toP[1,`]E (x). Secondly, since X/Xmis finite dimensional it follows thatlimk→∞kxnkkX/Xm =kxkX/Xm, and, thus, it follows that
kxk= lim
`→∞kP[1,`]E (x)k
= lim
`→∞ lim
k→∞kP[1,`]E (xnk)k
≤Klim sup
k→∞
kP[1,nE k](xnk)k
≤lim sup
k→∞ kxnkkX/Xm−Kε=kxkX/Xm−Kε, which is a contradiction sincekxk ≥ kxkX/Xm.
