GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS

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1 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS S. J. DILWORTH, MARIA GIRARDI, AND W. B. JOHNSON Studia Math. 140 (2000), no. 3, Abstract. A separable Banach space X contains l 1 isomorphically if and only if X has a bounded fundamental total wc 0-stable biorthogonal system. The dual of a separable Banach space X fails the Schur property if and only if X has a bounded fundamental total wc 0-biorthogonal system. 1. INTRODUCTION Generally it is easier to deal with Banach spaces that have some sort of basis structure, the most useful and commonly used structures being Schauder bases and finite-dimensional Schauder decompositions (FDD). Much research in Banach space theory has gone into proving that if a Banach space which has a Schauder basis or FDD possesses a certain property, then the space has a basis or FDD which reflects the property. While such theorems often give information (for example, by passing to suitable subspaces) about general spaces which do not have a basis or an FDD, they cannot give a classification of all separable spaces which have a certain property in terms of bases for the entire space unless the property itself implies the existence of a basis or FDD in a space which has the property. For that reason it is interesting to consider weaer structures than FDD s and Schauder bases which exist in every separable Banach space and try to prove that a separable Banach space has a certain property if and only if there is structure in the space which reflects the property. One useful basis-lie structure that has been considered for a long time is that of fundamental total biorthogonal system. Marushevich [M] showed in 1943 that each separable Banach space contains a fundamental total biorthogonal system. The main theorems of this paper characterize certain 1991 Mathematics Subject Classification. 46B99, 46B25, 46B20. Girardi is supported in part by NSF grant DMS and is a participant in the NSF Worshop in Linear Analysis and Probability, Texas A&M University (supported in part by NSF grant DMS ). Johnson is supported in part by NSF grant DMS , DMS , and by Texas Advanced Research Program under Grant No

2 2 DILWORTH, GIRARDI, JOHNSON geometric properties of a Banach space by which types of bounded fundamental total biorthogoal systems exist in the space. Theorem 1 shows that the dual of a separable Banach space X fails the Schur property if and only if X contains a bounded fundamental total wc 0-biorthogonal system. Recall that the dual of a Banach space X fails the Schur property if and only if X fails the Dunford-Pettis property or l 1 embeds into X. Theorem 2 shows that l 1 embeds in a separable Banach space X if and only if X contains a bounded fundamental total wc 0-stable biorthogonal system. Thirty-two years after Marushevich s result [M, 1943], Ovsepian and Pe lczyńsi showed [OP] that for each positive ε, each separable Banach space contains a [(1 + 2) 2 + ε]-bounded fundamental total biorthogonal system; the following year Pe lczyńsi [P] improved the bound to (1 + ε). The proofs of Theorems 1 and 2 use a combination of the methods in [OP] and [P]. Theorem 15 shows that if X is a separable Banach space containing l 1, then there is a [1 + 2+ε]-bounded fundamental biorthogonal system {x n,x n} in X X with the x n s arbitrarily close to an isomorphic copy of l 2 sitting in X. Section 5 shows that, in the statement of Theorem 15, the (1+ 2+ε) can not be replaced with ( ε). To the best of our nowledge, this is the first result in the literature which provides the existence of a bounded fundamental biorthogonal system in all spaces which have a certain property, and yet the bound for the systems cannot be arbitrarily close to one. 2. NOTATION and TERMINOLOGY Throughout this paper, X, Y, and Z denote arbitrary (infinite-dimensional real) Banach spaces. If X is a Banach space, then X is its dual space, B(X) is its (closed) unit ball, S(X) is its unit sphere, δ : X X is the natural point-evaluation isometric embedding, and x = δ(x). If Y is a subset of X, then sp{y } is the linear span of Y while [Y ] is the closed linear span of Y. Often used are the unit vector basis {δ n } of l 1, the Kronecer delta δ nm, and the space C(K) of continuous functions on a compact Hausdorff space K. If a > 0, then T L(X,Y) isanab isomorphic embedding provided a 1 x Tx b x for each x X; in this case, T o L(X,TX) denotes the bijective operator that agrees with T on X. A surjective τ isomorphic embedding T L(X,Y) is a τ isomorphism; in this case, X and Y are τ isomorphic. Recall that for a subset X of X and a subset Z of X

3 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 3 (1) X is fundamental if [X] =X, or, equivalently, the annihlator X of X in X is {0}, (2) Z is total if the wea -closure of sp{z} is X, or, equivalently, the preannihilator Z of Z in X is {0}, (3) for a fixed τ 1, Zτ-norms X (or X is τ-normed by Z) if z(x) x τ sup z Z\{0} z for each x X, (4) Z norms X if Z 1-norms X. If Zτ-norms X for a τ 1 then Z is total. Also, {x n,x n} n=1 in X Z is (1) a biorthogonal system if x n(x m )=δ nm, (2) M-bounded if {x n } and {x n} are bounded and sup n x n x n M, (3) bounded if it is M-bounded for some (finite) M, (4) fundamental if {x n } is fundamental, (5) total if {x n} is total. A biorthogonal system {x n,x n} n=1 in X X is: (1) a wc 0 -biorthogonal system if {x n} is a semi-normalized (i.e., bounded and bounded away from zero) wealy-null sequence, (2) a wc 0-stable biorthogonal system if, for each isomorphic embedding T of X into some Y, there exists a lifting {yn} of {x n} (i.e., T yn = x n for each n) such that {yn} is a semi-normalized wealy-null sequence in Y (or equivalently, such that {Tx n,yn}in Y Y is a wc 0 - biorthogonal system). Bases of type wc 0 were introduced in [FS] (cf. [S1, II.7 and pg ]). Recall that Z is injective if for each pair X and Y, each isomorphic embedding T L(X,Y), and each S L(X,Z), there exists S L(Y,Z) such that the following diagram commutes. T Y X S If Z is injective, then there exists λ 1 so that Z is λ-injective, i.e. S can be chosen so that S λ STo 1. Recall Z is a Grothendiec space if wea and wea sequential convergence in Z coincide; an injective space is S Z

4 4 DILWORTH, GIRARDI, JOHNSON a Grothendiec space (cf. [LT3, p. 188]). Z has the Schur property if wea and strong sequential convergence in Z coincide. All notation and terminology, not otherwise explained, are as in [DU] or [LT1]. 3. THE FINE LINE BETWEEN WC 0 AND WC 0 -STABLE The unit vectors {e p n,e q n}in l p l q, where 1 p< and q is the conjugate exponent of p, form a 1-bounded fundamental total wc 0 -biorthogonal system. For p = 1, they are even a wc 0-stable biorthogonal system, as the proof of (a) implies (b) in Theorem 2 shows. The next two theorems clarify the fine line between the existence of nice wc 0 -biorthogonal and wc 0 -stable biorthogonal systems. Theorem 1. The following statements are equivalent. (a) X fails the Schur property. (b) There is a bounded wc 0 -biorthogonal system in X X. And in the case that X is separable: (c) There is a bounded fundamental total wc 0-biorthogonal system {x n, x n} in X X. Furthermore for each ε>0:if(b) holds then the system can be taen to be (1+ε)-bounded; if (c) holds then the system can be taen to be [2(1+ 2) 2 +ε]- bounded and so that [x n] norms X. Recall (cf. [D2, p. 23]) that X fails the Schur property if and only if X fails the Dunford-Pettis property or l 1 X. Theorem 2. The following statements are equivalent. (a) l 1 X. (b) There is a bounded wc 0 -stable biorthogonal system in X X. And in the case that X is separable: (c) There is a bounded fundamental total wc 0-stable biorthogonal system {x n, x n} in X X. Furthermore for each ε>0:if(b) holds then the system can be taen to be (1+ε)-bounded; if (c) holds then the system can be taen to be [(1+ 2)+ε]- bounded and so that [x n](2+ε)-norms X. In this section are the proofs of the easier implications in the above theorems. The other implications follow from the results of the next section.

5 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 5 Proof of (b) implies (a) in Theorem 1. A wc 0-biorthogonal system in X X is enough to force X to fail the Schur property. Proof of (b) implies (a) in Theorem 2. Find an (isometric) embedding T of X into a C(K)-space. Assume that there is a wc 0-biorthogonal system {Tx n,yn}in C (K) C (K) with {x n } bounded, which would be the case if (b) held. If {x n } had a wealy Cauchy subsequence {x n }, then {Tx n } would be wealy Cauchy and {yn } would be wealy null, which cannot be since a C(K) space has the Dunford-Pettis property (cf. [D2, p. 20]). So, by Rosenthal s l 1 theorem, {x n } admits a subsequence that is equivalent to the unit vector basis of l 1. The above proof reveals somewhat more. Remar 3. In the definition of wc 0-stable biorthogonal system, if the word isomorphic is replaced with isometric then the statement of Theorem 2 remains true. Remar 4. If {x n,x n}in X X is either: 1. a bounded wc 0-biorthogonal system and X has the Dunford-Pettis Property or 2. a bounded wc 0-stable biorthogonal system, then each subsequence of {x n } contains a further subsequence that is equivalent to the unit vector basis of l 1. That (a) implies (b) in Theorem 1 (with the (1 + ε) bound) follows from Facts 5 7. Fact 5. Let {x n } n=1 be a wealy null sequence in X and {g n} be a bounded sequence in X and ε>0. Then there exists m N satisfying for infinitely many n N. x m,g n <ε This follows directly from the fact that, since {x n } n=1 is wealy null, there exists a finite sequence {λ m } N m=1 of positive numbers satisfying N ε max ±λ m x m < ± sup j g j m=1 and N m=1 λ m = 1 (cf. [W, p. 48, Exercise 13]).

6 6 DILWORTH, GIRARDI, JOHNSON Fact 6. Let {x n } n=1 be a normalized wealy null sequence in X and ε>0. Then there are a subsequence {x n } =1 and functionals {x n } =1 biorthogonal to {x n } =1 so that sup N x n < 1+ε. Proof. Fix a sequence {ε } =1 of positive numbers satisfying ε <ε/6. =1 Without loss of generality (pass to a subsequence), {x n } n=1 is a basic sequence with biorthogonal functional {f n } n=1 satisfying f n < 3. These f n s will be used to perturb functionals as needed. Without loss of generality (pass to a subsequence), there is a system {x n,g n } n=1 in X X satisfying g n < 1+ε/2 and x m,g n = δ mn when n m. To see how to find such a system by induction, consider a subsequence {n(j, )} =1 in N given at the beginning of the j th step (for the base step, let n(1,)=). Let x nj = x n(j,1) and find g nj in S (X ) satisfying g nj (x nj ) = 1. Find a subsequence {n(j +1,)} =1 of {n(j, )} =2 satisfying xn(j+1,), g nj <ε for each N and let g nj = g nj x n(j+1,), g nj f n(j+1,). =1 Without loss of generality (pass to a subsequence), x m,g n <ε m when m<n. To accomplish this, iterate Fact 5 to produce a sequence {{n(j, )} =1 } j=1 of sequences and a sequence { j } j=1 so that {n(j+1,)} =1 is a subsequence of {n(j, )} =1 and xn(j,j),g n(j+1,) < ε j. Then the subsequence n j = n(j, j ) wors. Clearly, the functionals x n := g n x m,g n f m {m N: m<n}

7 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 7 are biorthogonal to {x n } n=1 and are of norm at most 1 + ε. Fact 7. Let {x n, x n } n=1 be a biorthogonal system in X X with sup n x n < 1+ε for some ε>0and {x n} normalized and wea-star null. Then there is a subsequence {n } =1 along with a biorthogonal system { xn, x } n =1 in X X with sup x n < 1+ε. Proof. Without loss of generality (pass to a subsequence), there is a biorthogonal system {y n,x n} n=1 in X X with sup y n M<. n For just let X 0 be a separable subspace of X that 1-norms [x n] n=1 and tae a σ(x 0, X 0)-basic subsequence of {x n X0 } ([JR], cf. [D1, V.Exercise 7]). For each n N let E n := [x n ] F n := [x 1,...,x n]. Use the Principle of Local Reflexivity to find a sequence {z n } n=1 in X satisfying z n,x = x,x n = δ n when n and sup z n < 1+ε ε 0 n N for some ε 0 > 0. Fix a sequence {ε j } j=2 of positive numbers satisfying j=2 ε j < ε 0 M. Without loss of generality (pass to a subsequence), z n,x j <ε j when n<j. Clearly the vectors x n := z n z n,x j y j j=n+1 are biorthogonal to {x n} n=1 and are of norm at most 1 + ε. The next lemma provides a means by which to determine whether a wc 0 - biorthogonal system is wc 0 -stable.

8 8 DILWORTH, GIRARDI, JOHNSON Lemma 8. Let {x n, x n} be a biorthogonal system such that {x n} is a seminormalized wea -null sequence in X. Then {x n, x n} is a wc 0-stable biorthogonal system if and only if the operator S : X c 0 given by factors through an injective space. S(x) =(x n(x)) Proof. Let {x n,x n}be a biorthogonal system such that {x n} is a seminormalized wea -null sequence in X. First, assume that the above operator S factors through an injective space and let T : X Y be an isomorphic embedding. Consider the diagram T X S R L Y Z c 0 where Z is an injective space and S = LR. Since Z is injective, there exists R 1 L(Y,Z) such that the following diagram (totally) commutes. T X S R R 1 L Y Z c 0 Note that the operator S LR is given by (LR)(x) =(x n(x)) and so x n = R L (δ n ); similarly, the operator LR 1 has the form (LR 1 )(y) =(y n(y)) where y n = R 1 L (δ n ). It is easy to chec that {x n,x n}is indeed a wc 0-stable biorthogonal system: the commutativity of the diagram gives that T yn = x n, the wea-nullness of {yn} follows from the fact that Z is a Grothendiec space ({L δ n } is wea -null and thus wealy-null), and yn T 1 x n. Next assume that {x n,x n}is a wc 0-stable biorthogonal system. Find an embedding R from X into the injective space l (Γ) for some index set Γ. By the stability of the system, there exists a wealy-null sequence {yn} in l (Γ) such that R yn = x n. Define L: l (Γ) c 0 by L(f)=(yn(f)), for then S = LR.

9 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 9 The commutative diagram in the next proof was inspired by the Hagler Johnson proof [HJ] of the Josefson and Nissenzweig Theorem (cf. [D1, Chapter XII]). Con- Proof of (a) implies (b) in Theorem 2, along with the (1 + ε) bound. sider the following commutative diagram j i R 0 L X L l 1 c 0 where j is an isomorphic embedding, i is the formal injection, and ( ) R 0 (δ n )=r n and L(f) = fr n dµ for the Rademacher functions {r n }. Since L is 1-injective, there exists an operator R 2 such that the following diagram commutes n j i R 0 R 2 L X L and R 2 R0 jo 1. The operator S := LR 2 taes the form l 1 c 0 S(x) = (x n(x)) where x n = R 2 L (δ n ). It is easy to chec that {jδ n,x n}is a wc 0-stable biorthogonal system. Biorthogonality follows from the commutativity of the diagram. Since {δ n } is wea -null, so is {x n}. L is an injective space through which S factors. Furthermore, since R 0 and L both have norm one and 1 = x n(jδ n ), j 1 x n = R2L δ n R 2 L R 0 j 1 j 1 and so {x n} is semi-normalized. If l 1 embeds into X, then it (1 + ε)-embeds into X, thus one can arrange that sup n x n x n j j 1 1+ε. It is not difficult to see that, if X is any Banach space and ε>0, then there is a (2 + ε)-bounded biorthogonal system {x n,x n}in X X with {x n} wea -null. The first step towards this is the lemma below. o o o

10 10 DILWORTH, GIRARDI, JOHNSON Lemma 9. If X 0 is a finite co-dimensional subspace of X and ε>0, then there is a wea -closed finite co-dimensional subspace Y of X such that Y is (2 + ε)-normed by X 0. To see how to use Lemma 9 to produce the desired biorthogonal system {x n,x n} n=1, start with a normalized wea -null sequence {yn} n=1 in X (guaranteed to exist by the Josefson-Nissenzweig Theorem) and fix a sequence {ε n } n=1 of positive numbers tending to zero. Assume that { xj,x j} have been found. Let j<n X n = [ x j ] j<n and Z n =[x j ] j<n. By Lemma 9, there is a wea -closed finite co-dimensional subspace Y n of X that is (2 + ε/2)-normed by X n. Since Y n Z n is finite co-dimensional and wea -closed and {yn} n=1 is wea -null, there exists x n S (Y n Z n ) with x n y <εn n for some large n. Next find x n S (X n ) with 1 (2 + ε)x n ( x n ) and let x n := x n /x n ( x n ). Proof of Lemma 9. Let Y be the annihilator of any finite dimensional subspace of X that (1 + ε)-norms the annihilator of X 0. For then if f S(Y) then sup f(x 0 ) = x 0 S(X 0 ) inf f y y X 0 inf y X 0 max [ f y, sup x S(Y ) (f y )(x) ] max [ 1 y, 1 1+ε y ] inf y X 0 inf max [ 1 t, 0t< = (2+ε) 1. t 1+ε Lemma 9 is nearly best possible since, for each ε > 0, the one codimensional subspace X 0 of mean zero functions in L 1 does not (2 ε)-norm any finite co-dimensional subspace of L. Indeed, any finite co-dimensional subspace of L contains a norm one functional y that is bounded below by ε (just perturb a disjointly supported sequence of nonnegative norm one functions in L that are close to Y) and so y (x) 1 2 (1 + ε) for each x S (X 0 ). However, any one co-dimensional subspace X 0 of a Banach ]

11 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 11 space X does 2-norm a one co-dimensional subspace Y, namely Y := er P where P : X X 0 is a norm one projection. Indeed, if f S (Y) then f y 1 2 [ f y + P (f y ) ] for each y X 0. = 1 2 [ f y + y ] CONSTRUCTING FUNDAMENTAL TOTAL wc 0-BIORTHOGONAL SYSTEMS The constructions of fundamental total biorthogonal systems in the proofs of (a) implies (c) in Theorems 1 and 2 use the Haar matrices, which are summarized below. Remar 10. Fix m 0 and consider the 2 m -dimensional Hilbert space l 2m 2, along with its unit vector basis {e 2 j }2m j=1. The Haar basis {h m j }2m j=1 of l2m 2 can be described as follows. For 0 n m and 1 2 n let Thus I n = { j N : 2 m n ( 1) < j 2 m n }. I1 0 = {1, 2,..., 2 m } I1 1 = { 1,2,..., 2 m 1} and I2 1 = { 1+2 m 1,..., 2 m}. In general, the collection {I n}2n =1 of sets along the nth -level (disjointly) partitions {1, 2,...,2 m }into 2 n sets, each containing 2 m n consecutive integers, and I n is the disjoint union In = In In+1 2. Now let h m 1 = 2 m 2 j I 0 1 e 2 j and, for 0 n<mand 1 2 n, let h m 2 n + be supported on In as = 2n m 2 e 2 j. h m 2 n + j I n j I n+1 2 Note that {h m j }2m j=1 forms an orthonormal basis for l2m ( ) 2. Let H m = a m ij be the 2 m 2 m Haar matrix that transforms the unit vector basis of l 2m 2 onto the Haar basis; thus, the j th column vector of H m e 2 j

12 12 DILWORTH, GIRARDI, JOHNSON is just h m j and so H m is a unitary matrix. For example, H 2 = / / / /2 { } 2 m Let z j,zj be a biorthogonal sequence in S(X) j=1 X. Consider {x i,x i }2m i=1 where thus H m z 1. z 2 m = x i := 2 m j=1 x 1. x 2 m Since H m is a unitary matrix (H1) x i (x j)=δ ij (H2) [x i ] 2m i=1 =[z j] 2m and H m a m ij z j and x i := j=1 ] 2 m (H3) [x i ]2m i=1 [z = j. j=1 Note that, for each 1 i 2 m, (H4) a m i1 =2 m/2 z 1. and the l 1 -norm of the i th row of H m is bounded (H5) 2 m = m 2 m 1+ 2 and so j=1 a m ij z 2 m 2 m j=1 (H6) x i 2 m/2 z 1 + ( 1+ 2 ) max 1<j2 m z j (H7) x i 2 m/2 z1 + ( 1+ 2 ) max 1<j2 m zj (H8) for each x X = a m ij z j. x (x i ) x 2 m/2 z 1 + ( 1+ 2 ) max 1<j2 m x 1. x 2 m, x (z j ). Definition 11. A sequence {J } =1 is the disjoint union =1 J and of subsets of N is a blocing of N if N max J < min J +1

13 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 13 for each N. Given a blocing {J } =1 of N, let J 0 = {0} and J p := for each N. 0j< J j J o := J \{the first element in J } J po := J o N o := 0j< From the next theorem it easily follows, when X is separable, that (a) implies (c) in Theorem 1. Theorem 12. Let X fail the Schur property. Fix ε>0along with {a n, b n} in X X. Then there exists a [2(1 + 2) 2 + ε]-bounded wc 0 -biorthogonal system {x n, x n} in X X such that [a n ] [x n ] and [b n] [x n]. Proof. Without loss of generality, [a n ] n N and [b n] n N are each infinite dimensional. Fix a sequence {δ } =1 of positive numbers decreasing to zero. Since X fails the Schur property, there is a wealy-null sequence {wi } i=1 in S(X ). It suffices to find a system {x n,x n} n=1 in X X along with a blocing {J } =1 of N, a sequence {β n} n N o from (0, 2+ε], and an increasing sequence {i n } n N o from N, satisfying (1) x m(x n )=δ mn (2) x n ( 1+ 2 ) +ε (3) x n (2 + ε) ( 1+ 2 ) +ε (4) for each x S (X ), if n J then x (x n) δ + ( 1+ 2 ) ( ) ) max j J o x (β j w ij + βj δ (5) [a n ] n=1 [x n] n=1 (6) [b n] n=1 [x n] n=1. The construction will inductively produce blocs {x n,x n} n J. Let x 0 and x 0 be the zero vectors and j 0 = 0. Fix 1. Assume that {J j } 0j< along with {x n,x n} n J pand {i n } n J po and {β n} n J po have been constructed to satisfy conditions (1) through (4). Now to construct J along with {x n,x n} n J and {i n } n J o and {β n } n J o. Let =1 J o P := [x n] n J p and Q := [x n ] n J p

14 14 DILWORTH, GIRARDI, JOHNSON and n = max J p. The idea is to find a biorthogonal system {z n,zn} n J in P Q by first finding {z 1+n,z1+n } which helps guarantee condition (5) if is odd and condition (6) if even; however, {z 1+n,z1+n } would not necessarily satisfy conditions (2) through (4) and so J o and {z n,z n} n J o, along with {i n } n J o and {β n } n J o are constructed and then the Haar matrix is applied to {z n,zn} n J to produce {x n,x n} n J so that {x n,x n} n J p J with {i n } n J po and {β n} Jo n J po satisfy conditions (1) through (4). { Jo z1+n,z1+n } is constructed by a standard Gram-Schmidt biorthogonal procedure. If is odd, start in X. Let } = min {h: a h [x n ] nn h. Set z 1+n = a h nn x n(a h )x n, and for any y 1+n in X such that y 1+n (z 1+n ) 0, z1+n = y 1+n nn y1+n (x n )x n y1+n (z 1+n ) If is even, start in X. Let } = min {h: b h [x n] nn. h. Set z 1+n = b h nn b h (x n )x n, and, for any y 1+n in X such that z 1+n (y 1+n ) 0, z 1+n = y 1+n nn x n(y 1+n )x n z 1+n (y 1+n ). Clearly z 1+n (z 1+n )=1and z 1+n P and z 1+n Q.

15 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 15 Find a natural number m larger than one so that 2 m/2 max ( z 1+n, ) z 1+n < min ( ε, δ ) and let J := {1+n,...,2 m +n } and so J o := {2+n,...,2 m +n }. Let P := P [ z1+n ] and Q := Q [z 1+n ]. The next step is to find a biorthogonal system {z n,zn} n J o {i n } n J o and {β n } n J o satisfying ) ( ) {z n,zn} S ( P (2 + ε) B Q along with (1) and zn w i β n n < δ (2) for each n J o. Towards this, fix j J o and assume that a biorthogonal system {z n,zn} 2+n n<j along with {i n} 2+n n<j and {β n } 2+n n<j have been constructed so that conditions (1) and (2) hold for 2 +n n<j. Let X j := P [z n] 2+n n<j and Y j := Q [z n ] 2+n n<j. By Lemma 9, there is a wea -closed finite co-dimensional subspace Ỹj of X ) such that Ỹj is (2+ε/2)-normed by X j. Find i j >i j 1 and yj (Y S j Ỹj such that y j wi j <δ. Find z j S(X j ) such that 1 2+ε y j (z j ) := 1 β j and normalize zj := β j yj. This completes the inductive construction of {z n,zn} n J o along with the sets {i n } n J o and {β n } n J o. Now apply the Haar matrix to {z n,zn} n J to produce {x n,x n} n J. With help from the observations in Remar 10, note that {x n,x n} n J is biorthogonal and is in P Q. Furthermore, for each n in J, ( x n 2 m/2 z 1+n + 1+ ) 2 z j ε + ( 1+ ) 2 max j J o

16 16 DILWORTH, GIRARDI, JOHNSON and x n 2 m /2 z 1+n + (1+ ) 2 ε + (2+ε) ( 1+ ) 2 max j J o z j and for each x S (X ) x (x n) 2 m /2 z 1+n + (1+ ) 2 max x ( zj ) Thus δ + ( 1+ ) 2 max j J o j J o ( x ( β j w ij ) ) + βj δ. {x n,x n} n J p J with {i n } n J po and {β n} Jo n J po Jo is odd, then [a h ] hh [x n,z 1+n ] n J p while if is even, then [b h ] hh [ x n,z1+n ] satisfy conditions (1) through (4). If n J p [x n ] n J p J, [x n] n J p J. Clearly the constructed system {x n,x n} n=1, with the blocing {J } =1 of N and the increasing sequence {i n } n N o from N, and the sequence {β n } N o from (0, 2+ε], satisfy conditions (1) through (6). Some notation will be helpful in the next construction. Remar 13. Let X be a Banach space containing an isomorphic copy of l 1. Recall [P1, H2] that X contains an isomorphic copy of l 1 if and only if X contains an isomorphic copy of L 1. Thus X also contains an isomorphic copy of l 2. An isomorphic copy of L 1 (resp. l 2 )inx will be denoted by Z 1 (resp. Z 2 ). There is a norm on Z 2 which is equivalent to the usual norm on X and for which (Z 2, ) is Hilbertian; Z2 denotes Z 2 equipped with the new -norm. Since Z 2 is isometric to a Hilbert space, there is a unique inner product that induces its -norm; in Z 2, Hilbert space concepts are understood to be in Z 2. For example, a subset of Z 2 is orthonormal if, when viewed as a subset of Z 2, it is orthonormal in Z 2. A sequence {Y i } of finite-dimensional subspaces of Z 2 is an orthogonal finite-dimensional decomposition ( -fdd) provided Y i Y j for i j and each Y i is finite dimensional. Z 2 Y denotes the orthogonal complement of a subspace Y in Z 2.

17 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 17 Lemma 14. Let X be a separable Banach space containing an isomorphic copy of l 1 and ε>0. Then Z 1 can be taen so that a countable subset of it (2 + ε)-norms X. Proof. By [H1, DRT] there is a (1 + ε)-isomorphic copy of L 1 in X and so there is an embedding T : l 1 1 L 1 X satisfying, for each z l 1 1 L 1, z l1 1 L 1 Tz X (1 + ε) z l1 1 L 1. Moreover, the image {Tδ n }of the unit vector basis of l 1 can be assumed to be wea -null (since X is separable, {Tδ n } has a wea -convergent subsequence {Tδ n }, so just replace δ n by 1 2 (δ 2n δ 2n+1 ) ). Find a sequence {x n} n=1 in S(X ) such that {x n} n=n norms X for each N N. Fix β (0, 1) and let and Note that for each n N, y n = Tδ n +βx n Z 1 := [ {y n : n N} TL 1 ]. y n 1+ε+β. (3) The operator S : l 1 1 L 1 Z 1 defined by ( ) S αn δ n 1 f = α n yn + Tf illustrates that Z 1 is isomorphic to l 1 1 L 1. Indeed, fix αn δ n 1 f sp{δ n } n N 1 L 1. Then ( X S αn δ n 1 f) (1 + ε + β) α n + (1+ε) f L1 (1 + ε + β) α n δ n 1 f. l1 1 L 1 On the other hand, ( X T ( ) S αn δ n 1 f) = αn δ n 1 f + β α n x n X αn δ n 1 f β αn δ l1 n l1 1 L 1 (1 β) α n δ n 1 f. l1 1 L 1 ( ) Thus Z 1 is 1+β+ε 1 β -isomorphic to l 1 1 L 1, which is 3-isomorphic to L 1.

18 18 DILWORTH, GIRARDI, JOHNSON ( ) To see that {yn} n N is 1+ 1+ε β -norming for X, fixx S(X). Let δ>0 be such that δ(1 + δ) <βand find n N such that (Tδ n )(x) δ and 1 (1 + δ)x n(x), (4) for then by (3) and (4) yn(x) ( ) yn 1 β 1+ε+β 1+δ δ. Thus y sup n(x) n N yn β β +1+ε. So, for β sufficiently close to one, {yn} n N is (2 + 2ε)-norming for X. From Lemma 14 and Theorem 15 it easily follows, when X is separable, that (a) implies (c) in Theorem 2. Theorem 15. Let X be a separable Banach space containing l 1. From Remar 13, let Z 1 be total and Z 2 Z 1. Let {a n, b n} n=1 be in X Z 1 and fix ε, η > 0. Then there exists a [(1 + 2) + ε]-bounded wc 0 -stable biorthogonal system {x n, x n} n=1 in X X so that (15a) [a n ] n=1 [x n] n=1 (15b) [b n] n=1 [x n] n=1 Z 1 (15c) sup n N d(x n, Z 2 ) <η. In Section 5 it is shown that the [(1 + 2) + ε] can not be replaced with (1+ε) in Theorem 15. The following fact helps with the bound of the system in Theorem 15. It is due to Dvoretzy [Dv] and Milman [Mil]; a proof may be found in [P]. Fact 16. Let n, m, N be positive integers and δ>0. Then there is a positive integer K = K(n, m, N, δ) so that if (1) Y is a Banach space with K dim Y (2) E is a n-dimensional subspace of Y (3) H is a m-codimensional subspace of Y then there is a subspace F of H which is (1 + δ)-isomorphic to l N 2 and a projection P from E + F onto F with er P = E and P < 1+δ. In fact, they showed that P can be taen so that P I E+F < 1+δ. Proof of Theorem 15. The proof of Theorem 15 is similar to the proof of Theorem 12; thus, notation from the proof of Theorem 12 will be retained.

19 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 19 Fix a strictly decreasing sequence {η } =1 converging to zero with η 1 <η. It suffices to construct a system {x n,x n} n=1 in X X along with a blocing {J } =1 of N and a sequence {u n} n=1 from Z 2 satisfying (1) x m(x n )=δ mn (2) x n ( 1+ 2 ) +ε (3) x n 1+ε (4) x n u n η if n J (5) [u n] n J1 is orthogonal to [u n] n J2 for 1 2 (6) [a n ] n=1 [x n] n=1 (7) [b n] n=1 [x n] n=1 Z 1. Note that conditions (3) through (5) imply that {x n} n N is wealy-null in X. Clearly all that remains at this point is to show that the wc 0 -biorthogonal system {x n,x n} n=1 is indeed stable, which is done in the last step by using the condition that [x n] n=1 stays inside of Z 1. Let ε δ := The construction will inductively produce blocs {x n,x n} n J and {u n} n J. Fix 1. Assume that {J j } 0j< along with {x n,x n} n J p and {u n} n J p have been constructed to satisfy conditions (1) through (5). Now to construct J along with {x n,x n} n J and {u n} n J. The idea is to find a biorthogonal system {z n,zn} n J in P Q by first finding {z 1+n,z1+n } that helps guarantee condition (6) if is odd and condition (7) if even; however, {z 1+n,z1+n } would not necessarily satisfy conditions (2) and (3) and z1+n may be far from Z 2 and so J o and {z n,z n} n J o are then constructed and the Haar matrix is applied to {z n,zn} n J produce {x n,x n} n J and {u n} n J so that to {x n,x n} n J p J and {u n} n J p J satisfy conditions (1) through (5). Find {z 1+n,z1+n } just as in the proof of Theorem 12: in the case that is odd, be sure to choose y1+n in Z 1, which is possible since Z 1 is total. Find a natural number m larger than one so that 2 m /2 max ( z 1+n, z 1+n ) < min ( δ, η ).

20 20 DILWORTH, GIRARDI, JOHNSON Let [ ] E := {x n} n J p {z 1+n } ( ) H := Z 2 [u n] n J p [x n ] n J p Y := [Z 2 E ] N := 2 m 1 = J o. [z 1+n ] Use Fact 16 to find a subspace F of H, a projection P with ernel E, and a norm one isomorphism T so that E + F P F T l N 2 and Let {e n } n J o ( T 1 max ), P 2 < 1+δ. be an orthonormal basis for l N 2. For each n J o, let zn = T 1 e n and, using Local Reflexivity, find z n X that agrees, on E and F, with a norm-preserving Hahn-Banach extension of P T e n (E + F ) to X and satisfies z n < 1+δ P T e n. Then {z n,z n} n J o is a biorthogonal system in E F and max { z n : n J o } < 1+δ. Now apply the Haar matrix to {z n,zn} n J to produce {x n,x n} n J and let for each n in J. u n := j J o a m nj z j With help from the observations in Remar 10, note that for each n in J ( x n 2 m/2 z 1+n + 1+ ) 2 max z j j J o ( δ + 1+ ) 2 (1 + δ) ( 1+ ) 2 +ε

21 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 21 and x n 2 m /2 z 1+n + δ + (1+δ) 1 + 2δ 1 + ε. j J o j J o nj e j a m a m nj z j and since x n u n = a m n1 z n +1, x n u n = 2 m /2 z n +1 η. Thus, {x n,x n} n J p J and {u n} n J p J clearly satisfy conditions (1) through (5). This completes the inductive construction of the system {x n,x n} n=1 in X X, along with the blocing {J } of N and the sequence {u n} n=1 from Z 2, that satisfy conditions (1) through (7). The last step is to verify that {x n,x n} n=1 is indeed stable, which, by Lemma 8, is equivalent to verifying that the operator S : X c 0 given by S (x) =(x n(x)) factors through an injective space. Towards this, consider the following commutative diagram L 1 A j B l 1 X where j is an isomorphic embedding with range Z 1 and A and B are given by A(δ n )=jo 1 x n:= r n and B(δ n )=x n. Since A and B are of the form A (f) =(f( r n )) n and B (x )=(x (x n)) n, their ranges are contained in c 0 ; let A o and Bo be the corresponding maps with their ranges restricted to c 0. Thus the following diagram commutes.

22 22 DILWORTH, GIRARDI, JOHNSON j L A o B X o c 0 δ S X An appeal to Lemma 8 finishes the proof. Theorems 1 (c) is much easier to proof if one drops the total condition since then one can use the technique of Davis-Johnson-Singer ([DJ, Thm. 1] and [S2, Prop. 1]). As a partial illustration of this, we offer the following theorem, which gives a weaer result but a smaller constant than is provided by Theorems 1 (c). Theorem 17. Let X be a separable Banach space not containing l 1 such that X fails the Schur property. Fix ε>0. Then there is a (2 + ε)-bounded fundamental wc 0 -biorthogonal system {x n, x n} in X X. The meat in the proof of Theorem 17 is the following lemma. Lemma 18. Let X be a separable Banach space such that X fails the Schur property. Fix ε>0. Then there is a wc 0 -biorthogonal sequence {x n, x n} in S(X) X satisfying (1) x n 2+ε (2) {x n } is basic (3) [x n] +[x n ] is dense in X. Proof of Lemma 18. Fix a normalized wealy-null sequence {wn} in X,a dense sequence {d n } in X, a sequence {ε n } decreasing to zero, and a sequence {τ n } such that τ n > 1 and Πτ n <. It is sufficient to construct (a) a sequence {x n } n1 in S(X) (b) a sequence { x n} n1 in S(X ) (c) finite sets {F n } n0 in S(X ) with F 0 := (d) an increasing sequence { n } n=1 of integers that satisfy (4) x n Fn 1 {x i } i<n := X n (5) x n {d i } i<n {x i} i<n := Y n

23 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 23 (6) 1 2+ε x n(x n ) (7) x n w n εn (8) if x [x j ] jn, then there is f F n with x τ n f(x). For then just tae x n = x n/ x n(x n ). Note that (4) and (8) imply (2) while (5) and biorthogonality imply (3) since each d i has the form ( ) i i d i = d i x n(d i )x n + x n(d i )x n. n=1 The construction is by induction on n. To start, let x 1 = w 1. Find x 1 in S(X) that satisfies (6) and F 1 that satisfies (8). Fix n>1 and assume that the items in (a) through (d) have been constructed up through the (n 1) th -level. From this it is possible to find X n and Y n. By Lemma 9, there is a finite co-dimensional subspace Y of X that is (2 + ε 2 )-normed by X n. Find x n S (Y Y n ) along with n > n 1 such that (7) holds. Since Y is (2 + ε 2 )-normed by X n, there is x n S(X n ) such that 1 (2 + ε) x n(x n ). Now find F n satisfying (8). Proof of Theorem 17. First find the biorthogonal system {x n,x n}given by Lemma 18. The next step is to perturb this system to produce the desired system. Begin by finding a bijection p: N N N satisfying (i) {p(n, i)} i=1 is an increasing sequence ( for each n in N. Tae a dense set {y n } in B [x n] ). The underlying idea is to use {x p(n,i) } i to capture y n, along with {x p(n,i) } i, in the span of a small perturbation of {x p(n,i) } i. Towards this, with the help of (2) and the fact that {x p(n,i) } i is not equivalent to the unit vector basis of l 1, for each n find a sequence {a p(n,i) } i such that (ii) i a p(n,i) = (iii) i a p(n,i)x p(n,i) X. Let w p(n,i) := x p(n,i) ε ( ) sign a p(n,i) yn. Clearly {w n,x n}is a [(1 + ε)(2 + ε)]-bounded wc 0-biorthogonal system. n=1

24 24 DILWORTH, GIRARDI, JOHNSON Fix n 0 N and consider x [ w p(n0,i)]. For each m N, i ( m ) m x = εx (y n0 ). a p(n0,i)x p(n0,i) i=1 Combined with (ii) and (iii), this gives that x implies that x [ x p(n0,i)] i.thus [ {xp(n0 ],i)} i {y n 0 } i=1 [ ] w p(n0,i) i ap(n0,i) Combined with (3), it follows that {w n } n N is fundamental. [y n0 ], which in turn 5. BOUNDED FUNDAMENTAL BIORTHOGONAL SYSTEMS The nowledgeable reader notices that, in our proof of Theorem 15, a combination of the [OP] method (which produces [(1 + 2) 2 + ε]-bounded systems) and the [P] method (which produces (1 + ε)-bounded systems) is used to produce a (1 + 2+ε)-bounded system. Using just the [P] method in our proof of Theorem 15 will not guarantee that the x n s are in Z 1 nor close to Z 2. This difficulty is not purely technical. For indeed, consider the below special case of Theorem 15. Corollary 19. Fix ε, η > 0. Let Z 1 be a total subspace of l = l 1 that is isomorphic to L 1 and Z 2 be a subspace of Z 1 that is (1 + η)-isomorphic to l 2. Then there exists a [(1 + 2) + ε]-bounded fundamental biorthogonal system {x n, x n} n=1 in S(l 1) l satisfying sup n N d (x n, Z 2 ) <η. Lemma 20 shows that such subspaces Z 1 and Z 2 in Corollary 19 do exist. Corollary 24 shows that in Corollary 19, the [(1 + 2) + ε] can not be replaced with (1 + ε). However, if the requirement (15a) in the statement of Theorem 15 is removed (which would basically remove the fundamental condition), then the [P] method can be used to obtain this variant of Theorem 15 with (1 + ε) replacing [(1 + 2) + ε]. The proof of Lemma 14 gives that for each ε>0 there exists a strictly increasing surjective function i: (0, 1) (3(1 + ε), ) a strictly decreasing surjective function n: (0,1) (2 + ε, ) so that if X is a separable Banach space whose dual contains an isomorphic copy of L 1, then for each β (0, 1), there is a subspace Z 1 of X.

25 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 25 which is i(β) isomorphic to L 1 and which has a countable subset that n(β) norms X. However, if the dual space contains an isometric copy Z of L 1 then the above isomorphism constant i(β) can be improved. Lemma 20. There exists a strictly increasing surjective function i: (0, 1) (1, ) a strictly decreasing surjective function n: (0, 1) (2, ) so that if X is a separable Banach space whose dual contains an isometric copy of l 1 that is contractively complemented in some subspace Z of X, then for each β (0, 1), there is a subspace Z 1 of X which is i(β) isomorphic to Z and which has a countable subset that n(β) norms X. Since l contains an isometric copy Z of L 1, which in turn contains a contractively complemented subspace which is isometric to l 1, for each positive η, applying Lemma 20 with β sufficiently close to zero gives that there is a total subspace Z 1 of l that is (1 + η) isomorphic to L 1, which in turn contains a subspace Z 2 which is (1 + η) isomorphic to l 2. Proof of Lemma 20. Find {e n} n N in Z which is 1 equivalent to the standard unit vector basis of l 1 and a surjective contractive projection P : Z [e n] n N. Without loss of generality, {e n} n N is wea -null (similar to before, just replace {e n} with a wea -convergent subsequence { 1 2 (e 2n e 2n+1 )}, which will be 1 equivalent to {e n} and contractively complemented in [e n]). Find a sequence {x n} n=1 in S(X ) such that {x n} n=n norms X for each N N. Fix β (0, 1) and let and Note that for each n N, y n = e n + βx n Z 1 := [ {y n : n N} er P ]. Each element in Z has a unique expression as y n 1+β. (5) z = α n e n + f

26 26 DILWORTH, GIRARDI, JOHNSON where f er P ; the operator S : Z Z 1 defined by ( ) S αn e n + f = α n yn + f illustrates that Z 1 is isomorphic to Z. Indeed, fix αn e n + f sp{e n} n N +erp. Then z Sz = α n (e n yn) β α n = β P(z) β z. Thus (1 β) z Sz (1 + β) z for each z Z; thus, Z 1 is ( 1+β 1 β ) -isomorphic to Z. ( ) To see that {yn} n N is 1+ 1 β -norming for X, fixx S(X). Let δ>0 be such that δ(1 + δ) <βand find n N such that e n(x) δ and 1 (1 + δ)x n(x), (6) for then by (5) and (6) yn(x) yn 1 1+β Thus sup n N So {yn} n N is (1 + 1 β )-norming for X. ( β ) 1+δ δ yn(x) yn β β +1.. Recall that the modulus of convexity δ W :[0,2] [0, 1] of a Banach space W is { } δ W (ε) := inf 1 x + y 2 : x, y B(W) and x y ε and W is uniformly convex if δ W (ε) > 0 for each ε (0, 2]. If W is uniformly convex then δ W is a surjective continuous strictly increasing function (cf. [GK, pp ]). In a uniformly convex space, the midpoint of points near to the sphere that are far apart is uniformly bounded away from the sphere. This can be extended to convex combinations of points near to the sphere.

27 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 27 Lemma 21. Let W be a uniformly convex Banach space and ε and b be constants satisfying 0 < ε b 1. If {α j } j=1 R and {y j} j=1 B(W) satisfy α j =1 and α j y j > 1 ε, (7) j=1 j=1 then there is a finite subset F of N so that ε α j > 1 (8) 2b ε j F and for each j F, and α j 0. (sign α j) y j α j y j < δ 1 W (b) (9) j=1 Proof. Let {α j } j=1 R and {y j} j=1 B(W) satisfy (7). Set x 0 := α j y j. j=1 Without loss of generality each α j 0. Find x 0 S(W ) so that x 0 (x 0)= x 0 and let F = {j N : x 0(y j ) > 1 + ε 2b and α j 0}. The condition ε b guarantees that F is non-empty. Since 1 ε < α j x 0(y j ) + j x j F j Fα 0(y j ) (1 + ε 2b) 1 α j + j F α j j F = (1+ε 2b)+(2b ε) j Fα j, condition (8) holds. For each j F, 1 2 y j + x 0 > 1 ((1+ε 2b) + (1 ε) ) = 1 b 2 and so y j x 0 < δ 1 W (b)

28 28 DILWORTH, GIRARDI, JOHNSON by uniform convexity. Proposition 22. Let {x n, x n} n=1 be a biorthogonal system in S (l 1) l and W be a uniformly convex Banach space and Q L(l 1,W) be of norm at most one, all of which satisfy d (x n,q (KB(W ))) < 2η (10) for some constants K 1 and η>0.if 1 1 2η < 2 ( ) 1 4η K 3 δ W (11) 2K then {x n } n=1 is equivalent to the standard unit vector basis of l 1. More specifically if constants a and b satisfy ( η ) ( ) 1 4η a < b δ W (12) 2 K 2K then 3(b a) β n β 3b a n x n β n (13) n=1 n=1 n=1 for each {β n } n=1 in l 1. Note that if K = 1 and η = 0 then (11) becomes 0 < 2 ( ) 1 3 δ W ; 2 thus, if K is sufficiently close to 1 and η is sufficiently close to 0, as they often are in practice, then (11) does indeed hold. Proof. The underlying idea behind the proof is to use Lemma 21 to find a small perturbation { x n } n=1 of {x n} n=1 that are disjointly supported on the standard unit vector basis of l 1. For then, { x n } n=1 is equivalent to the standard unit vector basis of l 1 and so, for a small enough perturbation, {x n } n=1 is also equivalent to the standard unit vector basis of l 1. Find {wn} n=1 in KB(W ) so that Thus x n Q w n < 2η. (14) Qx n Qx n, wn 1 wn x n,x n x n,x n Q wn > 1 2η K

29 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 29 and so by the first inequality in (12) Write where j=1 α n j Qx n > 1 2a/3. x n = j=1 α n j δ j = 1 and let ε n j = sign αn j ; thus, Qx n = j=1 α n j Qδ j. So by Lemma 21 there is a sequence {F n } n=1 of finite subsets in N so that α n j a > 1 3b a j F n and for each j F n ε n j Qδ j Qx n < δ 1 W (b) and α n j 0. Let x n = αj n δ j. j F n To see that the F n s are disjoint, suppose that there is j 0 F n F m for some distinct n, m N. Pic τ { 1,1}so that τε m j 0 = ε n j 0. Then Qx n τqx m Qxn ε n j 0 Qδ j0 + τε m j0 Qδ j0 τqx m < 2δ 1 W (b). On the other hand, from (14) and the third inequality of (12) it follows that Qx n ± Qx m Q(x n ± x m ), wn 1 wn x n ± x m,x n x n ±x m,x n Q wn > 1 4η K 2δ 1 W (b). A contradiction, thus the finite subsets {F n } n=1 of N are indeed disjoint.

30 30 DILWORTH, GIRARDI, JOHNSON For each {β n } n=1 in l 1, β n β n x n n=1 n=1 β n x n β n x n x n n=1 n=1 ( 1 a ) β n 3b a and so (13) holds. = ( 1 2a 3b a n=1 ) β n n=1 ( ) a β n 3b a n=1 If W is uniformly convex then W is super-reflexive and so W has finite cotype; thus, there exists a cotype constant C q (W ) 1 for some q [2, ) so that ( n ) 1/q wi q C q (W ) n 2 avg θi =±1 θ i wi 1/2 (15) i=1 for each finite sequence {w i }n i=1 in W. Theorem 23. Let a uniformly convex Banach space W and ε 0 > 0 satisfy 1 1 = 2 ( ) 1+ε 0 3 δ 1 W. (16) 2(1 + ε 0 ) Let C q (W ) be as in (15) and 0 <ε 1 <ε 0. Then there exists a constant i=1 η = η (C q (W ),δ W,ε 1 )>0 (17) so that if (23a) {x n, x n} n=1 is a (1 + ε) bounded fundamental biorthogonal system in S (l 1 ) l (23b) Q L(l 1,W) (23c) Q is a (1 + η) isomorphic embedding (23d) d (x n,q W ) < η then ε>ε 1. The following notation helps crystallize condition (16) and simplify some technical arguments in the proof of Theorem 23.

31 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 31 Notation. Consider the functions l, u: [0, 1/4] [0, ) [0, ) [0, 1] given by 1 2η 1 l(η 1,η 2,ε) = 1 (1 + η 2 )(1 + ε) u(η 1,η 2,ε) = 2 ( ) (18) 3 δ 1 4η 1 W. 2(1 + η 2 )(1 + ε) Note that in each variable, l is a strictly increasing continuous function and u is a strictly decreasing continuous function. Condition (16) is equivalent to and l(0, 0,ε 0 ) = u(0, 0,ε 0 ) l(0, 0, ) : [0, ) onto [0, 1) u(0, 0, ) : [0, ) onto ( 0, 2 3 δ W ( )] 1 2 thus, for a uniformly convex space W, there is a unique ε 0 > 0 satisfying (16). Proof. The underlying idea behind the proof is that for sufficiently small η Proposition 22 gives that {x n } is equivalent to the standard unit vector basis of l 1 : indeed, condition (10) will hold and condition (16) implies (11). Then {x n} is equivalent to the standard unit vector basis of c 0. But if ε is small enough, then condition (23d) cannot hold since W has finite cotype. Let the hypotheses of Theorem 23 hold. Since l(0, 0,ε 1 )<l(0, 0,ε 0 )=u(0, 0,ε 0 )<u(0, 0,ε 1 ) there are constants a and b so that l(0, 0,ε 1 )<a<b<u(0, 0,ε 1 ). Find η = η (C q (W ),δ W,ε 1 )>0 sufficiently small enough so that l(η, η, ε 1 ) <a<b<u(η, η, ε 1 ) and so that there exists N N satisfying [ C (1 + η) 1+ 2a ] 3(b a) + 2Nη < N 1/q (19) where C = C q (W ). To see that condition (19) is easily accomplished, note that if [ ( 2C 3+ 2a )] q <N N 3(b a) and 0 <η 1 N then (19) holds. Let conditions (23a) through (23d) hold. Assume that ε ε 1. ;

32 32 DILWORTH, GIRARDI, JOHNSON By (23c), without loss of generality, for each w W, 1 1+η w Q w w. (20) Keeping with the notation from Proposition 22, let K =(1+η)(1+ε). From (23a), (23d), and (20) it follows that there is {wn} n=1 KB(W ) such that x n Q wn < 2η x n = Q wn. Thus (10) from Proposition 22 holds. Furthermore (11) also holds since l(η, η, ε) l(η, η, ε 1 ) <a<b<u(η, η, ε 1 ) u(η, η, ε). So by Proposition 22, since {x n } n=1 is fundamental, {x n} n=1 is equivalent to the standard unit vector basis of c 0 with 3(b a) β n x n sup β n β 3b a n n x n n F n=1 n=1 for each {β n } n=1 in c 0. For each finite subset F of N [ ] 1 wn q q C n avg θi =±1 θ i wi i=1 2 1/2 (21) The right-hand side of (21) mimics c 0 -growth in that for each choice {θ n } n F of signs θ n wn (1 + η) θ n Q w n n F n F [ ] (1 + η) θ n x n + θ n (x n Q w n) n F n F [ ] 3b a (1 + η) 3(b a) + 2 F η. The left-hand side of (21) mimics l q -growth since for each n N. Thus 1 x n = Q w n w n F 1 q C(1 + η) [ ] 3b a 3(b a) + 2 F η.

33 GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 33 This contradicts (19). Thus ε 1 <ε. Corollary 24. Let <ε 1 < 2+ 6 and, following the notation in (17), η = η (1,δ l2,ε 1 ). Let {x n, x n} n=1 be a (1 + ε) bounded fundamental biorthogonal system in S (l 1 ) l satisfying sup n N d (x n, Z 2 ) <η for some subspace Z 2 of l that is a (1 + η) isomorph of a Hilbert space. Then ε>ε 1. (22) Proof. Let W = l 2.ThusC 2 (W ) = 1. It is straight forward to verify that 147 ε 0 := 2+ 6 satisfies condition (16) of Theorem 23. Note that ε Let Z 2 be the predual of Z 2. There is an operator T L(l 1, Z 2 ) such that T L(Z 2,l ) is the formal pointwise embedding; for indeed, since Z 2 is reflexive, this formal pointwise embedding is wea -to-wea continuous. Similarly, by reflexivity, there is S L( Z 2,l 2 ) such that S L(l 2,Z 2 )isa (1 + η) isomorphism. Let Q = S T.Thus Q:l 1 T Z 2 S l2 and Q : l 2 S Z 2 T l and Q isa(1+η) isomorphic embedding. Thus condition (22) follows from Theorem 23. References [DJ] [DU] [D2] [D1] W. J. Davis and W. B. Johnson, On the existence of fundamental and total bounded biorthogonal systems in Banach spaces, Studia Math. 45 (1973), J. Diestel and J. J. Uhl, Jr., Vector measures, American Mathematical Society, Providence, R.I., 1977, With a foreword by B. J. Pettis, Mathematical Surveys, No. 15. Joe Diestel, A survey of results related to the Dunford-Pettis property, Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, N.C., 1979) (Providence, R.I.), Amer. Math. Soc., 1980, pp Joseph Diestel, Sequences and series in Banach spaces, Springer-Verlag, New Yor, 1984.

34 34 DILWORTH, GIRARDI, JOHNSON [DRT] Patric N. Dowling, Narcisse Randrianantoanina, and Barry Turett, Remars on James s distortion theorems, Bull. Austral. Math. Soc. 57 (1998), no. 1, [Dv] Aryeh Dvoretzy, Some results on convex bodies and Banach spaces, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Jerusalem Academic Press, Jerusalem, 1961, pp [E] Per Enflo, A counterexample to the approximation problem in Banach spaces, Acta Math. 130 (1973), [FS] Ciprian Foiaş and Ivan Singer, On bases in C([0, 1]) and L 1 ([0, 1]), Rev. Roumaine Math. Pures Appl. 10 (1965), [GK] Kazimierz Goebel and W. A. Kir, Topics in metric fixed point theory, Cambridge University Press, Cambridge, [H1] J. Hagler, Embeddings of l 1 into conjugate Banach spaces, Ph.D. dissertation, University of California, Bereley, CA, [HJ] J. Hagler and W. B. Johnson, On Banach spaces whose dual balls are not wea* sequentially compact, Israel J. Math. 28 (1977), no. 4, [H2] James Hagler, Some more Banach spaces which contain l 1, Studia Math. 46 (1973), [JR] W. B. Johnson and H. P. Rosenthal, On ω -basic sequences and their applications to the study of Banach spaces, Studia Math. 43 (1972), [LT3] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces, Springer-Verlag, Berlin, 1973, Lecture Notes in Mathematics, Vol [LT1] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer- Verlag, Berlin, 1977, Sequence spaces, Ergebnisse der Mathemati und ihrer Grenzgebiete, Vol. 92. [LT2] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Springer- Verlag, Berlin, 1979, Function spaces. [M] A. I. Marushevich, On a basis in the wide sense for liner spaces, Dol. Aad. Nau. 41 (1943), [Mil] V. D. Milman, Geometric theory of Banach spaces. II. Geometry of the unit ball, Uspehi Mat. Nau 26 (1971), no. 6(162), [OP] R. I. Ovsepian and A. Pe lczyńsi, On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in L 2, Studia Math. 54 (1975), no. 2, [P1] A. Pe lczyńsi, On Banach spaces containing L 1(µ), Studia Math. 30 (1968), [P] A. Pe lczyńsi, All separable Banach spaces admit for every ε>0fundamental total and bounded by 1 + ε biorthogonal sequences, Studia Math. 55 (1976), no. 3, [S1] Ivan Singer, Bases in Banach spaces. I, Springer-Verlag, New Yor, 1970, Die Grundlehren der mathematischen Wissenschaften, Band 154. [S2] Ivan Singer, On biorthogonal systems and total sequences of functionals, Math. Ann. 193 (1971), [W] P. Wojtaszczy, Banach spaces for analysts, Cambridge University Press, Cambridge, Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A. address: dilworth@math.sc.edu and girardi@math.sc.edu Department of Mathematics, Texas A&M University, College Station, TX 77843, U.S.A. address: johnson@math.tamu.edu