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On Weakly Null FDD’s in Banach Spaces

arXiv:math/9207207v1 [math.FA] 21 Jul 1992On Weakly Null FDD’s in Banach SpacesbyE. Odell,* H.P.

Rosenthal∗and Th. SchlumprechtAbstract.

In this paper we show that every sequence (Fn) of finite dimensionalsubspaces of a real or complex Banach space with increasing dimensions can be“refined” to yield an F.D.D. (Gn), still having increasing dimensions, so thateither every bounded sequence (xn), with xn ∈Gn for n ∈IN, is weakly null, orevery normalized sequence (xn), with xn ∈Gn for n ∈IN, is equivalent to theunit vector basis of ℓ1.Crucial to the proof are two stabilization results concerning Lipschitz func-tions on finite dimensional normed spaces.

These results also lead to other ap-plications. We show, for example, that every infinite dimensional Banach spaceX contains an F.D.D.

(Fn), with limn→∞dim(Fn) = ∞, so that all normalizedsequences (xn), with xn ∈Fn, n ∈IN, have the same spreading model over X.This spreading model must necessarily be 1-unconditional over X.§1. IntroductionLet (Fn) and (Gn) be two sequences of finite dimensional subspaces of a Banach space X.We say (Fn) is large if limn→∞dim Fn = ∞.

We say (Gn) is a refinement of (Fn) if thereis a strictly increasing sequence (kn) ⊂IN so that Gn is a subspace of Fkn for all n ∈IN. Ifeach (Fn) has a given basis bn = (f (n)i: 1 ≤i ≤dim Fn), we say (Gn) is a block refinementof (Fn) with respect to (bn) if Gn is spanned by a block basis of bn for all n. (Fn) is calledan F.D.D.

(Finite Dimensional Decomposition) if (Fn) is a Schauder-decomposition forits closed linear span. It is readily seen (using the standard Mazur argument) that everylarge sequence (Fn) has a large F.D.D.

refinement (Gn); moreover (Gn) can be chosen tobe a block-refinement of (Fn) with respect to (bn) for a given sequence of bases (bn) ofthe F.D.D. We say (Gn) is weakly null if every bounded sequence (xn) with (xn) ∈Gnfor all n, is weakly null.

We say (Gn) is uniformly-ℓ1 if there exists a C > 0 such thatall normalized sequences (xn) with xn ∈Gn for all n, are C-equivalent to the unit vector* Research partially supported by the National Science Foundation and TARP 235.1

basis of ℓ1. Of course (Gn) is uniformly-ℓ1 precisely when (Gn) is an ℓ1-F.D.D.

; that is,the closed linear span of the Gn’s is canonically isomorphic to (P ⊕Gn)1, the space of allsequences (gn) with gn ∈Gn for all n and ∥(gn)∥df= P ∥gn∥< ∞.Except as noted, our terminology is standard and may be found in the book [LT]. AllBanach spaces are assumed to be separable.If (xn) (resp.

(Gn)) is a (finite or infinite) sequence of elements of (resp. finite-dimensional subspaces of) a Banach space X, [xn] (resp.

[Gn]) denotes the closed linearspan of (xn) (resp. (Gn)).

SX denotes the unit sphere of X and Ba(X) its unit ball.Our main result is the following.Theorem 1. Let (Fn) be a large sequence of finite-dimensional subspaces of a Banachspace X.

Then there exists a large refinement (Gn) of (Fn) so that either (Gn) is a weaklynull FDD or (Gn) is an ℓ1-FDD. Furthermore if there is a given sequence (bn) of bases ofthe Fn’s with uniformly bounded basis constants, then the above sequence (Gn) can bechosen to be a block refinement of (Fn) with respect to (bn).Theorem 1 can be viewed as a block version of the ℓ1-theorem of the second namedauthor, which says that every normalized sequence (xn) in a Banach space X has a sub-sequence which is either equivalent to the unit vector basis of ℓ1 or is weak Cauchy [R1].Using Krivine’s theorem [K] (which is also used the in proof of Theorem 1), one gets fur-ther structural consequences of this block version.

Krivine’s theorem (as refined in [R2]and finally in [L]) may be formulated as follows:Given a large sequence (Fn) of finite-dimensional subspaces of a Banach spacewith bases (fn) with uniformly bounded basis constants, there exists a blockrefinement (Gn) of (Fn) with block bases (gn) of the fn’s so that for all n, n =dim(Gn) and gn is 1 + 1n-equivalent to the unit vector basis of ℓnp.Of course it thus follows that the Gn’s in the conclusion of Theorem 1 can be chosen tobe uniformly isomorphic to ℓnp, for some 1 ≤p ≤∞. We thus obtain immediately thefollowing result.Corollary 2.

Let (Fn) be a large sequence of finite dimensional subspaces of a Banachspace X, with given bases (bn) with uniformly bounded basic constants; and assume no2

normalized sequence (fn) with fn ∈Fn for all n, has a weak Cauchy subsequence. Thenthere exists 1 ≤p ≤∞and a block refinement (Gn) of (Fn) with respect to (bn), suchthat [Gn] is canonically isomorphic to (P ⊕ℓnp)1.Now Corollary 2 trivially implies that if X has the Schur property and contains ℓnp’suniformly, then (⊕ℓnp)1 embeds in X.

Of course this is trivial if 1 ≤p ≤2, since then ℓp isfinitely represented in ℓ1. However the following immediate block version does not appearto be obvious for any value of p larger than 1.Corollary 3.

Let X have the Schur property, and suppose, for some 1 < p ≤∞, thatℓp is block finitely represented in a particular basic sequence (xj) in X. Then some blockbasis of (xj) is equivalent to the natural basis of (P ⊕ℓnp)1.A famous question in Banach space theory was whether any infinite dimensional Ba-nach space X which does not contain ℓ1 isomorphically must contain an infinite-dimensionalsubspace with a separable dual.

This is equivalent to asking whether such an X containsa shrinking basic sequence (xn); i.e., a basic sequence (xn) so that each bounded blockbasis (yn) is weakly null. Of course if (xn) is such a sequence and (kn) is an increasingsequence in IN ∪{0} with kn+1 −kn →∞, then setting Fn = [xi]kn+1i=kn+1, (Fn) is a largeweakly null FDD.

However T. Gowers [G2] has recently solved the general problem inthe negative; i.e., there is a Banach space X not containing ℓ1, with no shrinking basicsequences. Nevertheless, Theorem 1 gives at once that every basic sequence in any X notcontaining ℓ1 has a block basis (xn) which yields large weakly null FDD’s as above.Corollary 4.

If ℓ1 is not isomorphically contained in X and (yn) is a basic sequence inX, then for each increasing sequence (kn) ⊂IN ∪{0} there exists a block basis (xn) of (yn)so that (Fn) is weakly null, where Fn = [xi]kn+1i=kn+1 for all n.Corollary 4 motivates the following problem.Problem. Assume ℓ1 is not contained in X.

Does there exist a basic sequence (xn) sothat all bounded “admissible” block bases of (xn) converge weakly to zero? (We call ablock basis (yn) of (xn) admissible if yn = Pℓni=1 α(n)ixm(n)iwhere for each n, ℓn ∈IN,3

(α(n)i) ∈IRℓn and ℓn ≤m(n)1< m(n)2< · · · < m(n)ℓn . In the terminology of [FJ] this justsays that for all n, supp(yn) (with respect to (xn)) is an admissible subset of IN.

)Another corollary of Theorem 1 is the following result, stated in [R6, Corollary 22] andproved there using Theorem 1 and the Borsuk antipodal mapping theorem. Corollary 5was obtained independently by W.B.

Johnson and T. Gamelin [CGJ].Corollary 5. Assume ℓ1 does not embed in X, where X is an infinite dimensional Banachspace.

Then there exists a normalized weakly null basic sequence (xi) in X possessing anormalized sequence of biorthogonal functionals.The main tools needed to prove Theorem 1 will be the following two finite dimensional“stabilization principles.” The first one was observed by V. Milman (see [MS, p.6]) inconnection with A. Dvoretzky’s famous theorem that in every infinite dimensional Banachspace one finds, for each ε > 0 and n ∈IN, an n-dimensional subspace F which is (1 + ε)-isomorphic to ℓn2. The second stabilization principle follows mainly from Lemberg’s [L]proof of Krivine’s theorem.First Stabilization Principle.For every C > 0, ε > 0 and k ∈IN there is an n = n(C, ε, k) ∈IN so that: If F is ann-dimensional normed space and f : F →IR is C-Lipschitz (i.e., |f(x) −f(y)| ≤C∥x −y∥for x, y ∈F), then there is a k-dimensional subspace G of F so thatoscf|SG≡sup|f(x) −f(y)| : x, y ∈SG< ε .Second Stabilization Principle.For all C > 0, ε > 0 and k ∈IN there is an n = n(C, ε, k) ∈IN so that if F is ann-dimensional normed space with a basis (xi)ni=1, whose basis constant does not exceed C,and if f : F →IR is C-Lipschitz, then there is a block basis (yi)ki=1 of (xi)ni=1 so thatoscf|S[yi]ki=1< ε .Since on the one hand the second stabilization principle nearly follows in a straightfor-ward manner from the proof of Krivine’s theorem (the only exception is the case F = ℓn∞),4

but on the other hand does not follow from the statement of Krivine’s theorem itself, wewill sketch the proof in section 3.The next result gives another application of the above stabilization principles. Theresult yields that for a given Lipschitz function f and large sequence (Fn) of X of finite-dimensional subspaces, there exists a large refinement (Gn), a Banach space E with aone-unconditional basis (ej), and a function ˜f : E →IR so that for all sequences (xi) withfi ∈SGi for all i, and all k, and all sequences (αi) ∈Ba(ℓ∞)˜f kXi=1αiei=limnk>···>n1→∞f kXi=1αixni.The result may be formulated quantitatively as follows: (c00 denotes the linear space offinitely supported real valued functions on IN.

We write for A, B ∈IR and ε > 0, Aε= Bif |A −B| < ε. )Theorem 6.

Let X be an infinite dimensional Banach space and let f : X →IR beLipschitz.Let (εn) ⊂IR+ with limn→∞εn = 0 and let (Fn) be a large sequence offinite dimensional subspaces of X. There exists a large refinement (Gn) of (Fn) and afunction ˜f : c00 ∩Ba(ℓ∞) →IR so that: For all k ∈IN and n1, n2, .

. ., nk ∈IN withk ≤n1 < n2 < · · · < nk, and all (αi)ki=1 ∈Ba(ℓk∞),˜f(α1, α2, .

. ., αk, 0, 0, .

. .

)εk= f kXi=1αixiwhenever xi ∈SGni for 1 ≤i ≤k. Moreover if each Fn has a given basis bn whose basisconstant does not exceed some fixed number, (Gn) may be chosen to be a block refinementof (Fn) with respect to (bn).Theorem 6 has a consequence concerning spreading models, and in fact the Banachspace “E” given in the above qualitative formulation may be chosen to be a spreadingmodel of X.

Recall that (see e.g., [BL] or [O]) every seminormalized basic sequence in Xadmits a subsequence (xn) satisfying: For all x ∈X, k ∈IN and (αi)ki=1 ⊆IR,limn1→∞limn2→∞. .

. limnk→∞x +kXi=1aixni exists.5

The limit is denoted by ∥x + Pki=1 aiei∥and defines a norm on X ⊕E where E = [ei]. Eis called a spreading model of X and X ⊕E is called a spreading model of (xi) over X.

(ei) is 1-unconditional over X if for all x ∈X, (αi)k1 ⊆IR and (εi) with |εi| = 1 for all i,x +kXi=1αiei =x +kXi=1εiαiei .Corollary 7. Every large sequence (Fn) of finite dimensional subspaces of an infinitedimensional Banach space X has a large refinement (Gn) with the following property: Allsequences (xn), with xn ∈SGn for n ∈IN, have the same spreading model E = [ei] overX.

In particular (ei) is 1-unconditional over X. Moreover, Gn can be chosen, so that forall ε > 0, k ∈IN and x ∈X there exists k0 ∈IN such that if k0 ≤n1 < n2 < · · · < nk, thenx +kXi=1αiei −x +kXi=1αixi < εwhenever xi ∈SGni , i = 1, .

. ., k, and (αi)ki=1 ∈Ba(ℓk∞).As usual, there is a corresponding “block refinement” version.

Corollary 7 follows fromTheorem 6 and a standard diagonal argument using the Lipschitz functions fx(y) = ∥x+y∥as x ranges over a dense subset of X. The result that every Banach space X has a spreadingmodel which is 1-unconditional over X is due to the second named author, see [R4], [R5].We note finally an application of Theorems 1 and 6 to the Banach-Saks property.

Thefollowing principle was discovered in 1975 (cf. [R2]; a proof may be found in [BL]).Given (xj) a semi-normalized weakly null sequence in a Banach space, there is asubsequence (x′j) so that either (x′j) has a spreading model isomorphic to ℓ1, or1n∥Pnj=1 x′′j ∥→0 as n →∞for all further subsequences (x′′j ) of (x′j).Now in fact one may assume in any case that (x′j) generates a spreading model, withbasis (bj) say; then the second alternative occurs precisely when (bj) itself is weakly null.In this case, one has ∥1nPnj=1 bj∥→0 as n →∞.

Then, e.g., setting εn = 2n∥Pnj=1 bj∥,(x′j) can be chosen so that 1n∥Pnj=1 x′′j ∥≤εn for all subsequences (x′′j ) of (x′j).The following result now follows immediately from Theorem 1 and Corollary 7.6

Corollary 8. Let (Fj) be a large sequence of finite dimensional subspaces of a Banachspace X, so that no normalized sequence (fj), with fj ∈Fj for all j, has a subsequenceequivalent to the ℓ1-basis.

Then there is a large weakly null FDD refinement (Gj) of (Fj),having one of the following mutually exclusive alternatives:1) (Gj) is uniformly anti-Banach-Saks; that is, there is a δ > 0 so thatlimn→∞1nnXj=1gj ≥δ limn→∞∥gn∥for all strictly increasing sequences (nj) in IN and all sequences (gj) ∈Q∞j=1 Ba Gnj.2) (Gj) is uniformly Banach-Saks; that is, there is a sequence (εj) of positive numberstending to zero so that1nnXj=1gj ≤εn for all n ,all strictly increasing sequences (nj) in IN, and all sequences (gj) ∈Q∞j=1 Ba Gnj. Moreoverif the Fn’s have bases bn with uniformly bounded basis constants, (Gn) may be chosen tobe a block refinement of (Fn) with respect to (bn).§2.

Proofs of Theorems 1 and 6.Proof of Theorem 1.Without loss of generality we can assume that X = C(K), thespace of all real or complex valued continuous functions on a compact metric space K.For f ∈C(K) we let f + = max(f, 0) in the real case; in the complex case we put f + =min((Re f)+, (Im f)+). For A ⊂K we let ∥· ∥A be the seminorm on C(K) defined by∥f∥A = supξ∈A |f(ξ)|.

Let (Fn) be a large sequence of finite dimensional subspaces ofC(K). Since (Fn) has a large FDD-refinement, we assume without loss of generality that(Fn) is already an FDD.We consider the following two cases.Case 1:(1) For all nonempty closed sets ˜K ⊂K, all ε > 0 and all large refinements (Hn) of (Fn)there is a relatively open set U ⊂˜K, U ̸= ∅, and a large refinement ( ˜Hn) of (Hn) sothatsup ∥f∥U < ε ,for f ∈[n∈INS ˜Hn .7

Case 2:(2) There are a nonempty closed set K0 ⊂K, ε0 > 0 and a large refinement (Hn) of(Fn) so that for all nonempty and relatively open sets U ⊂K0 and all further largerefinements ( ˜Hn) of (Hn),lim infn→∞suph∈S ˜Hn∥h∥U > ε0 .Clearly, cases 1 and 2 are mutually exclusive and the failure of one implies the otherholds. We will show that assuming case 1, we can find a weakly null large refinement (Gn)of (Fn).

Assuming case 2, we shall produce a uniformly-ℓ1 large refinement (Gn) of (Fn).Assume that (1) is satisfied and let ε > 0 be arbitrary. Let K(0) = K and (H(0)n ) =(Fn).We will choose by transfinite induction for each α < ω1 (where ω1 is the firstuncountable ordinal), a closed subset K(α) of K and a large refinement (H(α)n ) of (Fn), sothat(3) K(β) ⊆K(α) and, if K(α) ̸= ∅, then K(β) ⊄= K(α), whenever α < β.

(4) Except for perhaps finitely many elements, (H(β)n ) is a refinement of (H(α)n ) wheneverα < β. (5) For all ξ ∈K \ K(α),lim supn→∞supf∈SH(α)n|f(ξ)| ≤εAssume that for some α < ω1, (K(γ))γ<α and (H(γ)n )γ<α have been chosen.

If α = γ + 1and K(γ) = ∅set K(α) = ∅and (H(α)n ) = (H(γ)n ). If α = γ + 1 and K(γ) ̸= ∅, by (1) thereexists a large refinement (H(α)n ) of (H(γ)n ) and a relatively open set U ⊂K(γ), U ̸= ∅, sothat∥f∥U < ε for all f ∈[n∈INSH(γ)n.Set K(α) = K(γ) \ U.If α = limn→∞γn for some strictly increasing sequence (γn), set Kα = Tn∈IN Kγn andlet (H(α)n ) be a “diagonal sequence” of (H(γm)n)n,m∈IN, chosen such that for each m, exceptfor perhaps finitely many terms, (H(α)n ) is a large refinement of (H(γm)n).8

Since K is compact and metric, (thus K satisfies the Lindel¨offcondition) we concludethat for some α < ω1, K(β) = K(α) for α ≤β < ω1. By (3) it follows that K(α) = ∅andfrom (5) it follows that for all ξ ∈K,lim supn→∞supf∈SH(α)n|f(ξ)| ≤ε .We let (H(ε)n ) := (H(α)n ).

Repeating this argument for a sequence (εm) ⊂IR+ with εm ↓0one obtains for each m ∈IN, a large refinement (H(εm)n)n∈IN, of (H(εm−1)n), satisfying forall ξ ∈K,lim supn→∞supf∈SH(εm)n|f(ξ)| ≤εm .If we let (Gn) be a diagonal sequence of (H(εm)n)n,m∈IN, still satisfying limn→∞dim(Gn) =∞, we deduce that for all ξ ∈K,limn→∞supf∈SGn|f(ξ)| = 0 .Thus (Gn) is a weakly null large refinement of (Fn).We now assume that (2) is satisfied and let K0 ⊂K, ε0 > 0 and (Hn) be as in (2).Let ε1 = ε0 in the real case and ε1 = ε0/√2 in the complex case. Let D be a countabledense subset of K0.

By passing to a large refinement of (Hn) we can assume that(6)limn→∞supf∈SHn|f(ξ)| = 0 for all ξ ∈D .Indeed, let d1, d2, . .

. be an enumeration of D and let m1 < m2 < · · · be such thatdim Hmn ≥2n; then set H′n = {x ∈Hmn : x(di) = 0 for 1 ≤i ≤n}.

Now dim H′n ≥n forall n, so (H′n) is the desired large refinement. Let ε1/34 > δ > 0.

By induction we willchoose an increasing sequence of integers (kn) and for each n, a subspace Gn of Hkn anda finite set Πn consisting of nonempty relatively open subsets of K0 so that the followingconditions are satisfied:(7) dim(Gn) ≥n,and9

(8) For every g ∈SGn, and every U ∈Πn−1 (let Π0 = {K0}) there are U1, U2 ∈Πn,U1 ∪U2 ⊆U, so thatg+|U1 ≥ε1 −δand∥g∥U2 ≤δ .Once we have chosen (Gn) in this way we conclude that (Gn) must be uniformly-ℓ1. To seethis, fix (fn) with fn ∈SGn for all n ∈IN.

For each n, let An = {k ∈K : fn(k) > ε1 −δ}and Bn = {k ∈K : |fn(k)| < δ}. Evidently An ∩Bn = ∅for all n. We shall show that(An, Bn) is an independent sequence of pairs, in the terminology of [R1].

Once this is done,a refinement of the argument in [R1] yields that (fn) is 16ε1 -equivalent to the ℓ1-basis.Indeed, we first can inductively choose sets (U (n)i: i = 1, 2, . .

., 2n) ⊂Π(n) so thatf +n2n−1Si=1U(n)2i−1> ε1 −δand∥fn∥2n−1Si=1U(n)2i< δand so that U (n)2j∪U (n)2j−1 ⊂U (n−1)jfor n ∈IN and j = 1, 2, . .

., 2n−1.Now fix N, Iand J non-empty disjoint subsets of {1, . .

., N}, say with I ∪J = {1, . .

., N}. We see thatTn∈I An∩Tn∈J Bn is non-empty by defining the following sequence of sets C0, C1, .

. ., CN:Let U 01 = K0 = C0, 1 ≤n ≤N, and suppose Cn−1 is chosen with Cn−1 = U (n−1)jfor some1 ≤j ≤2n−1.

If n ∈I, set Cn = U (n)2j−1, otherwise set Cn = U (n)2j . Then the Cn’s satisfythat TNn=1 Cn ̸= ∅and for all n, Cn ⊂An if n ∈I, Cn ⊂Bn if n ∈J.Now let PNj=1 |aj| = 1 with aj = bj + icj for j ≤N.

By multiplying by −1, i or −i ifnecessary we may assume that PNj=1 b+j ≥1/4. Let I = {j ≤N : bj ≥0 and cj ≥0} andJ = {j ≤N : bj ≥0 and cj < 0}.

Thus eitherXj∈I(bj + cj) ≥18orXj∈J(bj −cj) ≥18 .Suppose the first sum exceeds 1/8. Now by the independence of (An, Bn), choose k ∈K10

such that f +j (k) > ε1 −δ for j ∈I and |fj(k)| < δ for j /∈I. Let fj(k) = Bj + iCj.

ThenNXj=1ajfj(k) ≥ Im NXj=1ajfj(k)=nXj=1(bjCj + Bjcj)≥Xj∈I(bjCj + Bjcj) −Xj /∈I|bjCj + Bjcj|≥(ε1 −δ)8−2δ > ε116 .A similar estimate ensues if the second sum exceeds 1/8. Thus (fn) is indeed 16ε1 -equivalentto the ℓ1-basis.Assume that for some n ≥1, Πn−1 and kn−1 (let k0 = 0) are chosen.

Now consider thefinite family of Lipschitz functions defined on C(K) by f 7→∥f +∥U, U ∈Πn−1. Since (Hn)is large, we may use the first stabilization principle in order to pass to a large refinement( ˜Hi) of (Hi)i>kn−1 so that for some family (a(U)i: U ∈Πn−1, i ∈IN) in IR+ we havea(U)i−δ4 < ∥f +∥U < a(U)i+ δ4whenever U ∈Πn−1, i ∈IN and f ∈S ˜Hi.

¿From (2) we deduce that there exists i0 ∈IN sothat for all i ≥i0 and U ∈Πn−1 we have a(U)i≥ε1 −δ4. Indeed, in the real case we onlyhave to observe that if ∥f∥U ≥ε0 then ∥f +∥U ≥ε0 or ∥(−f)+∥U ≥ε0; in the complex wefind for any f ∈C(K) for which ∥f∥U > ε0, a point ξ ∈U with |f(ξ)| > ε0 and then acomplex number a, with |a| = 1, so that Re(a · f(ξ)) = Im(a · f(ξ)) = (a · f(ξ))+.

Thus∥(a · f)+∥U ≥1√2∥f∥U > ε1. We deduce that(9)∥f +∥U > ε1 −δ2for all U ∈Πn−1, i ≥i0 and f ∈S ˜Hi.Now using (6), we pick, for each U ∈Πn−1, an element ξU ∈U ∩D and find an i1 ≥i0so that dim( ˜Hi1) ≥n and so that(10)supf∈S ˜Hi1|f(ξU)| < δ2 .11

Let (fs)ℓs=1 be a finite δ2-net for S ˜Hi1 . We find by (9) and (10) for each U ∈Πn−1, non-empty open subsets V (U)0, V (U)1, .

. .

, V (U)ℓso that f +s |V (U)s> ε1 −δ2 and ∥fs∥V (U)0<δ2,for s = 1, 2, . .

., ℓ.This implies that for all f ∈S ˜Hi1 we have ∥f∥V (U)0< δ, and forsome 1 ≤s ≤ℓ(namely the s for which ∥f −fs∥<δ2) we have f +|V (U)s> ε1 −δ.Set Πn =nV (U)0: U ∈Πn−1o∪nV (U)s: 1 ≤s ≤ℓ, U ∈Πn−1o, Gn = ˜Hi1, and choosekn > kn−1 so that ˜Hi1 ⊂Hkn. This completes the induction and thus the proof of thefirst version of Theorem 1.The “block-version” of Theorem 1 is proved in exactly the same way using the secondstabilization principle instead of the first.

One need only note that block refinements couldbe taken wherever we took simple refinements.Proof of Theorem 6. As in the proof of Theorem 1 we will only show the first version ofTheorem 6.

The “block-version” is left to the reader. We shall assume that X is a Banachspace over IR.

The complex case does not provide any further difficulties.Let f : X →IR be Lipschitz and let εn ↓0. We accomplish the proof by induction,insuring the conditions in the Theorem for a fixed k ≥2.

Precisely, we shall choose for eachk, a large sequence (G(k)n ) of finite dimensional subspaces so that (G(k+1)n) is a refinementof (G(k)n ) ((G(1)n ) ≡(Fn)), and a function C(k) : Ba(ℓk∞) →IR, so thatf(α1x1 + · · · + αkxk)εn1= C(k)(α1, . .

., αk)whenever (α1, . .

., αk) ∈Ba(ℓk∞) and xk ∈S(G(k)ni ), for all 1 ≤n1 < n2 < · · · < nk.Once this is done, then by diagonalization we finally find a large refinement (Gn) of(Fn) and functions C(k) : Ba(ℓk∞) →IR, k ∈IN so that for all k ∈IN and all k ≤n1

., αk)whenever xi ∈SGni , for i = 1, 2, . .

., k.Clearly we have thatC(k)(α1, α2, . .

., αk) = C(k+s)(α1, α2, . .

., αk, 0, 0, . .

., 0)12

for k, s ∈IN and (α1, α2, . .

., αk) ∈Ba(ℓk∞), and, thus, if we put˜f(α1, . .

. , αk, 0, 0, .

. .) = C(k)(α1, α2, .

. .

, αk) ,for k ∈IN and (αi)ki=1 ∈Ba(ℓk∞), ˜f has the required properties.We now indicate in detail how to carry this out for k = 2. First note the followingFact.

Let g : SX →IR be Lipschitz and let (Ln) be any large sequence of finite dimensionalsubspaces of X. Let δn ↓0.

There exist a large refinement (˜Ln) of (Ln) and C ∈IR suchthat for all n and y ∈S˜Ln,g(y)δn= C .This follows easily from the first stabilization theorem.One first obtains a largerefinement (˜˜Ln) of (Ln) and (Cn) ⊆IR such that g(y)δn/2=Cn for y ∈S˜˜Ln. (Cn) isbounded so for some subsequence (Ckn) and C ∈IR, |Ckn −C| < δn/2 for all n. Let˜Ln = ˜˜Lkn.Let H1 ≡F1.

Choose finite sets D1 ⊆D2 ⊆· · · ⊆Ba(ℓ2∞) and D1 ⊆D2 ⊆· · · ⊆SH1so that for all n, Dn is an εn-net for Ba(ℓ2∞) and Dn is an εn-net for SH1. For x ∈SH1and (α, β) ∈Ba(ℓ2∞), y 7→f(αx + βy) is a Lipschitz function on X.

Thus by iterating thefact above a finite number of times we obtain a large refinement (F (1,1)n)∞n=1 of (Fn) and(C(α, β, x))(α,β,x)∈D1×D1 ⊆IR such that for all (α, β) ∈D1, x ∈D1 and y ∈F (1,1)n,f(αx + βy)εn= C(α, β, x) .Repeating this argument inductively we obtain for all k ∈IN, a large refinement (F (1,k)n)∞n=1of (F (1,k−1)n)∞n=1 and (C(α, β, x))(α,β,x)∈Dk×Dk such thatf(αx + βy)εn= C(α, β, x)if (α, β) ∈Dk, x ∈Dk and y ∈F (1,k)n. By diagonalization we obtain a large refinement(F (1)n )∞n=1 of (Fn) with the propertyi) For k ∈IN, (α, β, x) ∈Dk × Dk, n ≥k, and y ∈F (1)n ,f(αx + βy)εn= C(α, β, x) .13

Suppose that the Lipschitz constant of f is K ≥1, i.e., |f(x) −f(y)| ≤K∥x −y∥. Thenfor (α, β), (α′, β′) ∈Ba(ℓ2∞), x, x′ ∈SH1, and ∥y∥= 1, we have|f(αx + βy) −f(α′x′ + β′y)| ≤K∥(αx −α′x) + (α′x −α′x′) + (β −β′)y∥≤K|α −α′| + |β −β′| + ∥x −x′∥.¿From this and i) we obtainii)|C(α, β, x) −C(α′, β′, x′)| ≤K|α −α′| + |β −β′| + ∥x −x′∥whenever (α, β), (α′, β′) ∈∪Dn and x, x′ ∈∪Dn.

Thus we can uniquely extend C(α, β, x)to a function C[1] : Ba(ℓ2∞) × SH1 →IR which satisfies ii) for all (α, β), (α′, β′) ∈Ba(ℓ2∞)and x, x′ ∈SH1. Furthermore, by replacing (F (1)n ) by an appropriate subsequence, we mayassume thatiii) For n ∈IN, (α, β, x) ∈Ba(ℓ2∞) × SH1 and y ∈SF (1)n ,f(αx + βy)εn= C[1](α, β, x) .Set H2 = F (1)n2 where n2 is chosen so that dim H2 > dim H1.

Proceeding as above weobtain a functionC[2] : Ba(ℓ2∞) × SH2 →IRand a large refinement (F (2)n ) of (F (1)n ) so thativ)|C[2](α, β, x) −C[2](α′, β′, x′)| ≤K|α −α′| + |β −β′| + ∥x −x′∥for all (α, β), (α′, β′) ∈Ba(ℓ2∞) and x, x′ ∈SH2 andv)f(αx + βy)εn= C[2](α, β, x)for all x ∈SH2, (α, β) ∈Ba(ℓ2∞) and y ∈SF (2)n .We continue in this manner obtaining a large refinement (Hn) of (Fn) and, for k ∈IN,functions C[k] : Ba(ℓ2∞) × SHk →IR satisfyingvi)|C[k](α, β, x) −C[k](α′, β′, x′)| ≤K|α −α′| + |β −β′| + ∥x −x′∥for (α, β), (α′, β′) ∈Ba(ℓ2∞) and x, x′ ∈SHk andvii)f(αx + βy)εn= C[k](α, β, x)for all (α, β) ∈Ba(ℓ2∞), x ∈SHk and y ∈SHn with n > k.14

(Actually it might be necessary to pass to a subsequence of (Hn) to obtain the preciseestimate vii). )We now apply the first stabilization result to finite sets of functions C[k](α, β, ·).

Letn ∈IN and 0 ≤¯ε ≤min{|α −α′| + |β −β′| : (α, β) ̸= (α′, β′) ∈Dn}.Consider fora fixed k the Lipschitz functions C[k](α, β, ·) : SHk →IR for (α, β) ∈Dn. If dim Hk issufficiently large there exists ˜Hk ⊆Hk, dim ˜Hk ≥n and (Ck(α, β))(α,β)∈Dn ⊆IR so thatC[k](α, β, x)¯ε= Ck(α, β) for all x ∈S ˜Hk and (α, β) ∈Dn.

Thus this plus vi) yields|Ck(α, β) −Ck(α′, β′)| ≤K|α −α′| + |β −β′|+ 2¯ε≤3K|α −α′| + |β −β′|for (α, β), (α′, β′) ∈Dj. The last inequality holds by the choice of ¯ε and the fact thatK ≥1.We inductively use this argument for the parameters (n, ¯εn) where ¯εn ↓0 rapidlychosen depending on (Dn) and (εn).We obtain a large refinement (In) of (Hn) withdim In ≥n and functions Cn : Dn →IR satisfyingviii)|Cn(α, β) −Cn(α′, β′)| ≤3K|α −α′| + |β −β′|for (α, β), (α′, β′) ∈Dnandix) For all x ∈SIn and (α, β) ∈Dn,f(αx + βy)εn= Cn(α, β) whenever y ∈SIq ,q > n .For n ∈IN, the function (Ck|Dn)k≥n are uniformly Lipschitz.

Thus by a compactnessargument we can find a Lipschitz function C(2) : ∪Dj →IR and k1 < k2 < · · · so that forall n and (α, β) ∈Dn,C(2)(α, β)εn= Ckn(α, β) .C(2) thus uniquely extends to a continuous function C(2) : Ba(ℓ2∞) →IR. Letting (G(2)n )∞n=1be a suitable subsequence of (Ikn) we obtainx)f(αxn1 + βxn2)εn1= C(2)(α, β) for all (α, β) ∈Ba(ℓ2∞),n1 < n2, xn1 ∈SG(2)n1 and xn2 ∈SG(2)n2which was what was needed to be proved in the case k = 2.15

§3. A Sketch of the Proof of the Second Stabilization PrincipleThe reader unfamiliar with Lemberg’s proof might first wish to read that argument (see[MS, Ch.12]).In order to shorten the proof we will not only use Lemberg’s proof ofKrivine’s theorem but also the quantitative version of this theorem.Theorem 9.

(see [R3])For every C > 1, ε > 0 and k ∈IN, there is an n = n(C, ε, k) ∈INso that: If F is a Banach space of dimension n and if (fi)ni=1 is a basis of F having basisconstant not exceeding C, then there exists a block basis (gi)ki=1 of (fi)ni=1 and a p ∈[1, ∞]so that (gi)ki=1 is (1 + ε)-equivalent to the unit basis of ℓkp.In view of Theorem 9 we only have to prove the second stabilization principle for finitedimensional ℓp-spaces. Using a compactness argument, similar to the argument of [R3] bywhich Theorem 9 was deduced from the finite dimensional version of Krivine’s theorem,we only have to show the following claim.Claim 1.

Let X = ℓp, 1 ≤p < ∞, or X = c0, and let f : X →IR be a Lipschitz function.For each ε > 0 and k ∈IN there exists a block basis (yi)ki=1 of (ei) (the unit vector basisof X) so that osc(f|S[yi]ki=1 ) < ε.Proof of Claim 1. We need some notation.

For x, y ∈c00 we say x and y have the samedistribution, and write xdist= y, if x = Pki=1 αieni and y = Pki=1 αiemi for some k ∈IN,(αi)ki=1 ⊂IR, resp.IC, and n1 < n2 < · · · nk and m1 < m2 < · · · mk. We define forx, y ∈X ∩c00,dis(x, y) = infn∥¯x −¯y∥: ¯xdist= x and ¯ydist= yo.For x ∈c00, we let supp(x) = {i ∈IN : xi ̸= 0}, and write x < y for x, y ∈c00 ifmax(supp(x)) < min(supp(y)).We first reduce claim 1 to the case that f is 1-unconditional and 1-spreading.

Bythis we mean that f(P αiei) = f(P |αi|eni) for all P αiei ∈X and all strictly increasingsequences (ni) ⊂IN.In order to reduce claim 1 to the 1-unconditional and 1-spreading case we first passto a sequence ni ⊂IN for whichf (1) ℓXi=1αiei= limk1→∞limk2→∞. .

. limkℓ→∞f ℓXi=1αienki16

exists for all ℓand scalars α1, α2, . .

., αℓ. It follows that f (1) is 1-spreading on X.

If X isdefined over IR we let ℓn = 2, for n ∈IN, and put (ξi)ℓni=1 = (1, −1). If X is defined overIC we let ℓn = n, for n ∈IN, and (ξj)ℓnj=1 = e(i2πj/n).

If X = ℓp, 1 ≤p < ∞, let (un) be asequence in X with u1 < u2 < · · · andundist=1(nℓn)1/pnXs=1ℓnXt=1ξte(s−1)ℓn+t ,for n ∈IN .If X = c0 we let (un) be a sequence in X, with u1 < u2 < · · ·, andundist=nXs=1sn ·ℓnXt=1ξte(s−1)ℓn+t +n−1Xs=1n −snℓnXt=1ξte(n+s−1)ℓn+tNote that (un) is normalized, and that from the fact that f (1) is 1-spreading it follows thatfor some sequence εn ↓0 and some subsequence (˜un) of (un),f (1) kXj=1αj ˜uj+nεn= f (1) kXj=1σjαj ˜uj+nwhenever k, n ∈IN, |σj| = 1, for j = 1, 2, . .

., k, and ∥Pkj=1 αjej∥= 1.Pass now to a subsequence (ni) ⊂IN for whichf (2) ℓXi=1αiei≡limk1→∞limk2→∞. .

. limkℓ→∞f (1) ℓXi=1αi˜unkiexists, whenever ℓ∈IN and (αi)ℓi=1 ∈c00.

f (2) is 1-unconditional and 1-spreading and weneed only prove that claim 1 is true for f (2). Thus, in order to finish the proof of claim 1we need to show the following claim 2.Claim 2.

For every ε > 0 and k ∈IN there is a block basis (xi)ki=1 of (ei) which isnormalized in X, having the property that the setB+(x1, . .

., xk) =( kXi=1αixi : 0 ≤αi ≤1 ,kXi=1αixi = 1)has diameter less than ε with respect to dis(·, ·).17

Proof of Claim 2.Case 1: X = ℓp, 1 ≤p < ∞.In this case we consider as in [L] the “rationalized” version of ℓp, i.e., ℓp(D) ={(xq)q∈D : Pq∈D |xq|p < ∞} where D = IQ ∩(0, 1). Let (eq)q∈D denote the natural basisof ℓp(D).

For n ∈IN define the operator Tn : ℓp(D) →ℓp(D byTnXq∈Dαqeq=nXj=1Xq∈Dαqe(q+j−1)/n .For every n ∈IN, λn = n1/p is an approximate eigenvalue of Tn [L] and since Tn and Tmcommute for n, m ∈IN one can choose for a fixed m ∈IN, m ≫k, and δ > 0 a vectoru = Pq∈D uqeq ∈Ba(ℓp(D)) so that supp(u) = {q ∈D : uq ̸= 0} is finite, and so that∥Tn(u) −n1/pu∥< δ for all m ≤n.Let x1 < x2 < · · · < xm be elements of ℓp (= ℓp(IN)), each having the same distributionas u (i.e., xkdist= Psi=1 uqiei where q1 < q2 < · · · < qs and supp(u) = {q1, q2, . .

., qs}). Wededuce that for any scalars α1, α2, .

. ., αk with ∥Pki=1 αiei∥= 1 we havedisx1 ,kXi=1αixiδ1= disx1 ,kXi=1mim1/pxiδ≤dis1m1/pmXi=1xi ,kXi=1mim1/pxi≤1m1/pkXi=1dis miXj=1xj , (mi)1/pxi≤k · δ ,where m1, .

. ., mk ∈IN with Pki=1 mi = m are chosen so that P |αi −( mim )1/p| is minimal,and where δ1 depends on m and decreases to zero for m →∞.

Thus, choosing m bigenough and δ small enough we deduce claim 2 in the case that X = ℓp.Case 2: X = c0In this case Lemberg’s argument does not work, but we are able to explicitly writedown the desired vectors x1, x2, . .

., xk.For 0 < r < 1 we will define a sequence of vectors (y(n) : n ∈IN0) in Ba(c0) ∩c00. We18

put y(0) = e1 and assuming y(n) = Pℓni=1 y(n)iei is chosen we puty(n+1) =ℓnXi=1rn+1e3(i−1)+1 + y(n)ie3(i−1)+2 + rn+1e3(i−1)+3(thus y(1) = (r, 1, r, 0, . .

. ), y(2) = (r2, r, r2, r2, 1, r2, r2, r, r2, 0, .

. .

), etc.). Choosing r < 1big enough and n ∈IN big enough and letting x1 < x2 < · · · < xk all have the samedistribution as y(n) one also deduces claim 2.Remark.

For the case X = c0, T. Gowers [G] independently obtained a deeper version ofclaim 1. He showed that for every Lipschitz function f : c0 →IR and every ε > 0 there isan infinite dimensional subspace Y of c0 so that osc(f|SY ) < ε.

This is false for X = ℓp(1 ≤p < ∞) [OS].References[BL]B. Beauzamy and J.-T. Laprest´e, Mod`eles ´etal´es des espaces de Banach, Travauxen Cours, Herman, Paris, 1984. [CGJ] B.J.

Cole and T. Gamelin and W.B. Johnson, Analytic Disks in Fibers over the unitball of a Banach space, Mich. J.

Math. (to appear).[G1]T.

Gowers, Lipschitz functions on classical spaces, preprint.[G2]T. Gowers, A space not containing c0, ℓ1 or a reflexive subspace, preprint.[K]J.L.

Krivine, Sous-espaces de dimension finie des espaces de Banach reticul´es, Ann.of Math. 104 (1976), 1–29.[L]H.

Lemberg, Nouvelle d´emonstration d’un th´eor`eme de J.L. Krivine sur la finierepr´esen-tation de ℓp dans un espace de Banach, Israel J.

Math. 39 (1981), 341–348.[LT]J.

Lindenstrauss and L. Tzafriri, “Classical Banach Spaces I,” Springer-Verlag, NewYork, 1975.[MS]V.D. Milman and G. Schechtman, “Asymptotic Theory of Finite DimensionalNormed Spaces,” LNM 1200, Springer-Verlag, New York, 1986.[O]E.

Odell, Applications of Ramsey theorems to Banach space theory, in “Notes inBanach Spaces” (H.E. Lacey, ed.

), University Press, Austin and London, 379–404,1980.[OS]E. Odell and Th.

Schlumprecht, The Distortion Problem, preprint.[R1]H. Rosenthal, A characterization of Banach spaces containing ℓ1, Proc.

Nat. Acad.Sci.

USA 71 (1974), 2411–2413.[R2]H. Rosenthal, Weakly independent sequences and the Banach-Saks property, Ab-stract for the 1975 Durham Symposium, Bull.

London Math. Soc.

8 (1976), 22–24.[R3]H. Rosenthal, On a theorem of J.L.

Krivine concerning block finite-representabilityof ℓp in general Banach spaces, J. Funct. Anal.

28 (1978), 197–225.19

[R4]H. Rosenthal, Some remarks concerning unconditional basic sequences, LonghornNotes (1982-83), The University of Texas at Austin, 15–48.[R5]H. Rosenthal, The unconditional basis sequence problem, Contemp.

Math. 52 (1986),70–88.[R6]H.

Rosenthal, Some aspects of the subspace structure of infinite dimensional Ba-nach spaces, in “Approximation Theory and Functional Analysis” (C.K. Chui, ed.

),Academic Press, San Diego, CA, 1991, 151–176.July 9, 199220

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