GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS

S. J. DILWORTH, MARIA GIRARDI, 119911AND W. B. JOHNSON

Studia Math. 140 (2000), no. 3, 243–271

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 weaker 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-like structure that has been considered for a long time is that of fundamental total biorthogonal system. Markushevich [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-9622841 and is a participant in the NSF Workshop in Linear Analysis and Probability, Texas A&M University (supported in part by NSF grant DMS-9311902). Johnson is supported in part by NSF grant DMS-9623260, DMS-9900185, and by Texas Advanced Research Program under Grant No. 010366-163.

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 Markushevich’s result [M, 1943], Ovsepian and Pełczyński showed [OP] that for each positive ε, each separable Banach space contains a [(1 + \sqrt{2})^{2} + ε]-bounded fundamental total biorthogonal system; the following year Pełczyński [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 + \sqrt{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+\sqrt{2}+ε) can not be replaced with (1.02 + ε). To the best of our knowledge, 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 x̂ = 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 Kronecker 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) is an ab–isomorphic embedding provided

a^{−1} ∥x∥ ⩽ ∥T x∥ ⩽ b ∥x∥

for each x ∈ 𝔛; in this case, T_{o} ∈ L(𝔛, T 𝔛) 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^{∗}

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 weak^{∗}-closure of sp{Z} is 𝔛^{∗}, or, equivalently, the preannihilator Z^{⊤} of Z in 𝔛 is {0},

(3) for a fixed τ ⩾ 1, Z τ-norms X (or X is τ-normed by Z) if

∥x∥ ⩽ τ sup_{z∈Z\\{0\}} \frac{z(x)}{∥z∥}

for each x ∈ X,

(4) Z norms X if Z 1-norms X. If Z τ-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) weakly-null sequence,

(2) a wc_{0}^{∗}-stable biorthogonal system if, for each isomorphic embedding T of 𝔛 into some Y, there exists a lifting {y_{n}^{∗}} of {x_{n}^{∗}} (i.e., T^{∗}y_{n}^{∗} = x_{n}^{∗} for each n) such that {y_{n}^{∗}} is a semi-normalized weakly-null sequence in Y^{∗} (or equivalently, such that {T x_{n}, y_{n}^{∗}} in Y × Y^{∗} is a wc_{0}^{∗}- biorthogonal system).

Bases of type wc_{0}^{∗} were introduced in [FS] (cf. [S1, II.7 and pg. 625–626]).

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 S̃ ∈ L(Y, Z) such that the following diagram commutes.

[figure: 𝔛; S; Z; T; Y; S̃]

If Z is injective, then there exists λ ⩾ 1 so that Z is λ-injective, i.e. S̃ can be chosen so that ∥S̃∥ ⩽ λ ∥ST_{o}^{−1}∥. Recall Z is a Grothendieck space if weak^{∗} and weak sequential convergence in Z^{∗} coincide; an injective space is

4 DILWORTH, GIRARDI, JOHNSON

a Grothendieck space (cf. [LT3, p. 188]). Z has the Schur property if weak 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 W C_{0}^{∗} AND W C_{0}^{∗}-STABLE

The unit vectors {e_{n}^{p}, e_{n}^{q}} 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 taken to be (1+ε)-bounded; if (c) holds then the system can be taken to be [2(1+\sqrt{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 taken to be (1+ε)-bounded; if (c) holds then the system can be taken to be [(1+\sqrt{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.

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 {T x_{n}, y_{n}^{∗}} in C (K) × C^{∗} (K) with {x_{n}} bounded, which would be the case if (b) held. If {x_{n}} had a weakly Cauchy subsequence {x_{n_{k}}}, then {T x_{n_{k}}} would be weakly Cauchy and {y_{n_{k}}^{∗}} would be weakly 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.

Remark 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. ■

Remark 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 weakly null sequence in 𝔛 and {g_{n}} be a bounded sequence in 𝔛^{∗} and ε > 0. Then there exists m ∈ ℕ satisfying

|⟨x_{m}, g_{n}⟩| < ε

for infinitely many n ∈ ℕ.

This follows directly from the fact that, since {x_{n}}_{n=1}^{∞} is weakly null, there exists a finite sequence {λ_{m}}_{m=1}^{N} of positive numbers satisfying

max_{±} ∥∑_{m=1}^{N} ±λ_{m}x_{m}∥ < \frac{ε}{sup_{j} ∥g_{j}∥}

and ∑_{m=1}^{N} λ_{m} = 1 (cf. [W, p. 48, Exercise 13]).

6 DILWORTH, GIRARDI, JOHNSON

Fact 6. Let {x_{n}}_{n=1}^{∞} be a normalized weakly null sequence in 𝔛 and ε > 0. Then there are a subsequence {x_{n_{k}}}_{k=1}^{∞} and functionals {x_{n_{k}}^{∗}}_{k=1}^{∞} biorthogonal to {x_{n_{k}}}_{k=1}^{∞} so that sup_{k∈ℕ} ∥x_{n_{k}}^{∗} ∥ < 1 + ε.

Proof. Fix a sequence {ε_{k}}_{k=1}^{∞} of positive numbers satisfying

∑_{k=1}^{∞} ε_{k} < ε/6 .

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, k)}_{k=1}^{∞}

in ℕ given at the beginning of the j^{th} step (for the base step, let n(1, k) = k). Let x_{n_{j}} = x_{n(j,1)} and find g̃_{n_{j}} in S (𝔛^{∗}) satisfying g̃_{n_{j}}(x_{n_{j}}) = 1. Find a subsequence {n(j + 1, k)}_{k=1}^{∞} of {n(j, k)}_{k=2}^{∞} satisfying

|⟨x_{n(j+1,k)}, g̃_{n_{j}}⟩| < ε_{k}

for each k ∈ ℕ and let

g_{n_{j}} = g̃_{n_{j}} − ∑_{k=1}^{∞}⟨x_{n(j+1,k)}, g̃_{n_{j}}⟩ f_{n(j+1,k)} .

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, k)}_{k=1}^{∞}}_{j=1}^{∞}

of sequences and a sequence {k_{j}}_{j=1}^{∞} so that {n(j+1, k)}_{k=1}^{∞} is a subsequence of {n(j, k)}_{k=1}^{∞} and

|⟨x_{n(j,k_{j})}, g_{n(j+1,k)}⟩| < ε_{j} .

Then the subsequence n_{j} = n(j, k_{j}) works.

Clearly, the functionals

x_{n}^{∗} := g_{n} − ∑_{{m∈ℕ: m<n}}⟨x_{m}, g_{n}⟩ f_{m}

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 > 0 and {x_{n}^{∗}} normalized and weak-star null. Then there is a subsequence {n_{k}}_{k=1}^{∞} along with a biorthogonal system {x_{n_{k}}, x_{n_{k}}^{∗}}_{k=1}^{∞} in 𝔛 × 𝔛^{∗} with sup_{k} ∥x_{n_{k}}∥ < 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_{n} ∥y_{n}∥ ⩽ M < ∞ .

For just let 𝔛_{0} be a separable subspace of 𝔛 that 1-norms [x_{n}^{∗}]_{n=1}^{∞} and take a σ(𝔛_{0}^{∗}, 𝔛_{0})-basic subsequence of {x_{n}^{∗}|_{𝔛_{0}}} ([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_{k}^{∗}⟩ = ⟨x_{k}^{∗}, x_{n}^{∗∗}⟩ = δ_{nk} when k ⩽ n

and

sup_{n∈ℕ}∥z_{n}∥ < 1 + ε − ε_{0}

for some ε_{0} > 0. Fix a sequence {ε_{j}}_{j=2}^{∞} of positive numbers satisfying

∑_{j=2}^{∞} ε_{j} < \frac{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} − ∑_{j=n+1}^{∞}⟨z_{n}, x_{j}^{∗}⟩ y_{j}

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 DILWORTH, GIRARDI, JOHNSON

Lemma 8. Let {x_{n}, x_{n}^{∗}} be a biorthogonal system such that {x_{n}^{∗}} is a seminormalized weak^{∗}-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

S(x) = (x_{n}^{∗} (x))

factors through an injective space.

Proof. Let {x_{n}, x_{n}^{∗}} be a biorthogonal system such that {x_{n}^{∗}} is a seminormalized weak^{∗}-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

[figure: Y; Z; L; c_{0}; R; 𝔛; S; T]

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.

[figure: Y; Z; L; c_{0}; R; 𝔛; S; T; R_{1}]

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 check that {x_{n}, x_{n}^{∗}} is indeed a wc_{0}^{∗}-stable biorthogonal system: the commutativity of the diagram gives that T^{∗}y_{n}^{∗} = x_{n}^{∗}, the weak-nullness of {y_{n}^{∗}} follows from the fact that Z is a Grothendieck space ({L^{∗}δ_{n}} is weak^{∗}-null and thus weakly-null), and ∥y_{n}^{∗}∥ ⩾ ∥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 weakly-null sequence {y_{n}^{∗}} in ℓ_{∞}^{∗}(Γ) such that R^{∗}y_{n}^{∗} = x_{n}^{∗}. Define L: ℓ_{∞}(Γ) → c_{0} by L(f) = (y_{n}^{∗} (f)), for then S = LR. ■

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]).

Proof of (a) implies (b) in Theorem 2, along with the (1 + ε) bound. Consider the following commutative diagram

[figure: 𝔛; L_{∞}; L; c_{0}; R_{0}; ℓ_{1}; i; j]

where j is an isomorphic embedding, i is the formal injection, and

R_{0}(δ_{n}) = r_{n}  and  L(f) = (∫ fr_{n} dμ)_{n}

for the Rademacher functions {r_{n}}. Since L_{∞} is 1-injective, there exists an operator R_{2} such that the following diagram commutes

[figure: 𝔛; L_{∞}; L; c_{0}; R_{0}; ℓ_{1}; i; j; R_{2}]

and ∥R_{2}∥ ⩽ ∥R_{0}j_{o}^{−1}∥.

The operator S := L R_{2} takes the form

S(x) = (x_{n}^{∗} (x)) where x_{n}^{∗} = R_{2}^{∗} L^{∗} (δ_{n}) .

It is easy to check that {jδ_{n}, x_{n}^{∗}} is a wc_{0}^{∗}-stable biorthogonal system. Biorthogonality follows from the commutativity of the diagram. Since {δ_{n}} is weak^{∗}-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}^{∗}∥ = ∥R_{2}^{∗}L^{∗}δ_{n}∥ ⩽ ∥R_{2}∥ ∥L∥ ⩽ ∥R_{0}j_{o}^{−1}∥ ⩽ ∥j_{o}^{−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_{o}^{−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}^{∗}} weak^{∗}-null. The first step towards this is the lemma below.

10 DILWORTH, GIRARDI, JOHNSON

Lemma 9. If 𝔛_{0} is a finite co-dimensional subspace of 𝔛 and ε > 0, then there is a weak^{∗}-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 weak^{∗}-null sequence {y_{n}^{∗}}_{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

{x_{j}, x_{j}^{∗}}_{j<n}

have been found. Let

𝔛_{n} = [x_{j}^{∗}]_{j<n}^{⊤} and Z_{n} = [x_{j}]_{j<n}^{⊥} .

By Lemma 9, there is a weak^{∗}-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 weak^{∗}-closed and {y_{n}^{∗}}_{n=1}^{∞} is weak^{∗}-null, there exists x_{n}^{∗} ∈ S (Y_{n} ∩ Z_{n}) with ∥x_{n}^{∗} − y_{k_{n}}^{∗}∥ < ε_{n} for some large k_{n}. Next find x̃_{n} ∈ S (𝔛_{n}) with 1 ⩽

(2 + ε)x_{n}^{∗} \binom{}{x_{n}} and let x_{n} := x̃_{n}/x_{n}^{∗} \binom{}{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_{x_{0}∈S(𝔛_{0})} |f(x_{0})| = inf_{y^{∗}∈𝔛_{0}^{⊥}} ∥f − y^{∗}∥

⩾ inf_{y^{∗}∈𝔛_{0}^{⊥}} max [ ∥f∥ − ∥y^{∗}∥ , sup_{x∈S(Y^{⊤})} |(f − y^{∗}) (x)| ]

⩾ inf_{y^{∗}∈𝔛_{0}^{⊥}} max [1 − ∥y^{∗}∥ , \frac{1}{1+ε} ∥y^{∗}∥ ]

⩾ inf_{0⩽t<∞} max [1 − t , \frac{t}{1+ε}]

= (2 + ε)^{−1} .

■

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) ⩽ \frac{1}{2} (1 + ε) for each x ∈ S (𝔛_{0}). However, any one co-dimensional subspace 𝔛_{0} of a Banach space 𝔛 does 2-norm a one co-dimensional subspace Y, namely Y := ker P where P: 𝔛^{∗} → 𝔛_{0}^{⊥} is a norm one projection. Indeed, if f ∈ S (Y) then

∥f − y^{∗}∥ ⩾ \frac{1}{2} [ ∥f − y^{∗}∥ + ∥P (f − y^{∗})∥ ]

= \frac{1}{2} [ ∥f − y^{∗}∥ + ∥y^{∗}∥ ] ⩾ \frac{1}{2}

for each y^{∗} ∈ 𝔛_{0}^{⊥}.

4. 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.

Remark 10. Fix m ⩾ 0 and consider the 2^{m}-dimensional Hilbert space ℓ_{2}^{2^{m}}, along with its unit vector basis {e_{j}^{2}}_{j=1}^{2^{m}}.

The Haar basis {h_{j}^{m}}_{j=1}^{2^{m}} of ℓ_{2}^{2^{m}} can be described as follows. For 0 ⩽ n ⩽ m and 1 ⩽ k ⩽ 2^{n} let

I_{k}^{n} = {j ∈ ℕ : 2^{m−n} (k − 1) < j ⩽ 2^{m−n} k} .

Thus

I_{1}^{0} = {1, 2 , …, 2^{m}}
I_{1}^{1} = {1, 2 , …, 2^{m−1}} and I_{2}^{1} = {1 + 2^{m−1} , …, 2^{m}} .

In general, the collection {I_{k}^{n}}_{k=1}^{2^{n}} of sets along the n^{th}-level (disjointly) partitions {1, 2, …, 2^{m}} into 2^{n} sets, each containing 2^{m−n} consecutive integers, and I_{k}^{n} is the disjoint union I_{k}^{n} = I_{2k−1}^{n+1} ∪ I_{2k}^{n+1}. Now let

h_{1}^{m} = 2^{\frac{−m}{2}} ∑_{j∈I_{1}^{0}} e_{j}^{2}

and, for 0 ⩽ n < m and 1 ⩽ k ⩽ 2^{n}, let h_{2^{n}+k}^{m} be supported on I_{k}^{n} as

h_{2^{n}+k}^{m} = 2^{\frac{n−m}{2}} [ ∑_{j∈I_{2k−1}^{n+1}} e_{j}^{2} − ∑_{j∈I_{2k}^{n+1}} e_{j}^{2}] .

Note that {h_{j}^{m}}_{j=1}^{2^{m}} forms an orthonormal basis for ℓ_{2}^{2^{m}}.

Let H_{m} = (a_{ij}^{m}) be the 2^{m} × 2^{m} Haar matrix that transforms the unit vector basis of ℓ_{2}^{2^{m}} onto the Haar basis; thus, the j^{th} column vector of H_{m} is just h_{j}^{m} and so H_{m} is a unitary matrix. For example,

H_{2} = [\begin{matrix}2^{−1} & +2^{−1} & +2^{−1/2} & 0 \\ 2^{−1} & +2^{−1} & −2^{−1/2} & 0 \\ 2^{−1} & −2^{−1} & 0 & +2^{−1/2} \\ 2^{−1} & −2^{−1} & 0 & −2^{−1/2} \end{matrix}] .

Let {z_{j}, z_{j}^{∗}}_{j=1}^{2^{m}} be a biorthogonal sequence in S(𝔛) × 𝔛^{∗}. Consider {x_{i}, x_{i}^{∗}}_{i=1}^{2^{m}} where

H_{m} [ z_{1} ; ⋮ ; z_{2^{m}} ] = [ x_{1} ; ⋮ ; x_{2^{m}} ]  and  H_{m} [ z_{1}^{∗} ; ⋮ ; z_{2^{m}}^{∗}] = [ x_{1}^{∗} ; ⋮ ; x_{2^{m}}^{∗}] ,

thus

x_{i} := ∑_{j=1}^{2^{m}} a_{ij}^{m}z_{j}  and  x_{i}^{∗} := ∑_{j=1}^{2^{m}} a_{ij}^{m}z_{j}^{∗} .

Since H_{m} is a unitary matrix (H1) x_{i}^{∗}(x_{j}) = δ_{ij} (H2) [x_{i}]_{i=1}^{2^{m}} = [z_{j}]_{j=1}^{2^{m}} (H3) [x_{i}^{∗}]_{i=1}^{2^{m}} = [z_{j}^{∗}]_{j=1}^{2^{m}}.

Note that, for each 1 ⩽ i ⩽ 2^{m},

(H4) a_{i1}^{m} = 2^{−m/2} and the ℓ_{1}-norm of the i^{th} row of H_{m} is bounded

(H5) ∑_{j=1}^{2^{m}} |a_{ij}^{m}| = 1 + \sqrt{2} − 2^{\frac{1−m}{2}} ^{m→∞}↗ 1 + \sqrt{2} and so

(H6) ∥x_{i}∥ ⩽ 2^{−m/2} ∥z_{1}∥ + (1 + \sqrt{2}) max_{1<j⩽2^{m}} ∥z_{j}∥ (H7) ∥x_{i}^{∗}∥ ⩽ 2^{−m/2} ∥z_{1}^{∗}∥ + (1 + \sqrt{2}) max_{1<j⩽2^{m}} ∥z_{j}^{∗}∥ (H8) for each x^{∗∗} ∈ 𝔛^{∗∗}

|x^{∗∗}(x_{i}^{∗})| ⩽ ∥x^{∗∗}∥ 2^{−m/2} ∥z_{1}^{∗}∥ + (1 + \sqrt{2}) max_{1<j⩽2^{m}} |x^{∗∗} (z_{j}^{∗})| .

Definition 11. A sequence {J_{k}}_{k=1}^{∞} of subsets of ℕ is a blocking of ℕ if ℕ is the disjoint union ∪_{k=1}^{∞}J_{k} and

max J_{k} < min J_{k+1}

GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 13

for each k ∈ ℕ. Given a blocking {J_{k}}_{k=1}^{∞} of ℕ, let J_{0} = {0} and

J_{k}^{p} := ⋃_{0⩽j<k} J_{j}

J_{k}^{o} := J_{k} \ {the first element in J_{k}}

J_{k}^{po} := ⋃_{0⩽j<k} J_{k}^{o}

ℕ^{o} := ⋃_{k=1}^{∞} J_{k}^{o}

for each k ∈ ℕ.

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 > 0 along with {a_{n}, b_{n}^{∗}} in 𝔛 × 𝔛^{∗}. Then there exists a [2(1 + \sqrt{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_{k}}_{k=1}^{∞} of positive numbers decreasing to zero. Since 𝔛^{∗} fails the Schur property, there is a weakly-null sequence {w_{i}^{∗}}_{i=1}^{∞} in S(𝔛^{∗}).

It suffices to find a system {x_{n}, x_{n}^{∗}}_{n=1}^{∞} in 𝔛 × 𝔛^{∗} along with a blocking {J_{k}}_{k=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 + \sqrt{2}) + ε

(3) ∥x_{n}^{∗}∥ ⩽ (2 + ε) (1 + \sqrt{2}) + ε

(4) for each x^{∗∗} ∈ S (X^{∗∗}), if n ∈ J_{k} then |x^{∗∗} (x_{n}^{∗})| ⩽ δ_{k} + (1 + \sqrt{2}) max_{j∈J_{k}^{o}} ( |x^{∗∗} (β_{j}w_{i_{j}}^{∗})| + β_{j}δ_{k} )

(5) [a_{n}]_{n=1}^{∞} ⊂ [x_{n}]_{n=1}^{∞}

(6) [b_{n}^{∗}]_{n=1}^{∞} ⊂ [x_{n}^{∗}]_{n=1}^{∞} .

The construction will inductively produce blocks {x_{n}, x_{n}^{∗}}_{n∈J_{k}}. Let x_{0} and x_{0}^{∗} be the zero vectors and j_{0} = 0. Fix k ⩾ 1. Assume that {J_{j}}_{0⩽j<k} along with {x_{n}, x_{n}^{∗}}_{n∈J_{k}^{p}} and {i_{n}}_{n∈J_{k}^{po}} and {β_{n}}_{n∈J_{k}^{po}} have been constructed to satisfy conditions (1) through (4). Now to construct J_{k} along with {x_{n}, x_{n}^{∗}}_{n∈J_{k}} and {i_{n}}_{n∈J_{k}^{o}} and {β_{n}}_{n∈J_{k}^{o}}.

Let

P_{k} := [x_{n}^{∗}]_{n∈J_{k}^{p}}^{⊤}  and  Q_{k} := [x_{n}]_{n∈J_{k}^{p}}^{⊥}

14 DILWORTH, GIRARDI, JOHNSON

and

n_{k} = max J_{k}^{p} .

The idea is to find a biorthogonal system {z_{n}, z_{n}^{∗}}_{n∈J_{k}} in P_{k} × Q_{k} by first finding {z_{1+n_{k}}, z_{1+n_{k}}^{∗}} which helps guarantee condition (5) if k is odd and condition (6) if k even; however, {z_{1+n_{k}}, z_{1+n_{k}}^{∗}} would not necessarily satisfy conditions (2) through (4) and so J_{k}^{o} and

{z_{n}, z_{n}^{∗}}_{n∈J_{k}^{o}} ,

along with {i_{n}}_{n∈J_{k}^{o}} and {β_{n}}_{n∈J_{k}^{o}} are constructed and then the Haar matrix is applied to {z_{n}, z_{n}^{∗}}_{n∈J_{k}} to produce {x_{n}, x_{n}^{∗}}_{n∈J_{k}} so that

{x_{n}, x_{n}^{∗}}_{n∈J_{k}^{p} ∪J_{k}}

with {i_{n}}_{n∈J_{k}^{po}∪J_{k}^{o}} and {β_{n}}_{n∈J_{k}^{po}∪J_{k}^{o}} satisfy conditions (1) through (4).

{z_{1+n_{k}}, z_{1+n_{k}}^{∗}} is constructed by a standard Gram-Schmidt biorthogonal procedure. If k is odd, start in 𝔛. Let

h_{k} = min {h: a_{h} ∉ [x_{n}]_{n⩽n_{k}}} .

Set

z_{1+n_{k}} = a_{h_{k}} − ∑_{n⩽n_{k}} x_{n}^{∗}(a_{h_{k}})x_{n} ,

and for any y_{1+n_{k}}^{∗} in 𝔛^{∗} such that y_{1+n_{k}}^{∗}(z_{1+n_{k}}) ≠ 0,

z_{1+n_{k}}^{∗} = \frac{y_{1+n_{k}}^{∗} − ∑_{n⩽n_{k}} y_{1+n_{k}}^{∗}(x_{n})x_{n}^{∗}}{y_{1+n_{k}}^{∗}(z_{1+n_{k}})} .

If k is even, start in 𝔛^{∗}. Let

h_{k} = min {h: b_{h}^{∗} ∉ [x_{n}^{∗}]_{n⩽n_{k}}} .

Set

z_{1+n_{k}}^{∗} = b_{h_{k}}^{∗} − ∑_{n⩽n_{k}} b_{h_{k}}^{∗}(x_{n})x_{n}^{∗} ,

and, for any y_{1+n_{k}} in 𝔛 such that z_{1+n_{k}}^{∗}(y_{1+n_{k}}) ≠ 0,

z_{1+n_{k}} = \frac{y_{1+n_{k}} − ∑_{n⩽n_{k}} x_{n}^{∗}(y_{1+n_{k}})x_{n}}{z_{1+n_{k}}^{∗}(y_{1+n_{k}})} .

Clearly z_{1+n_{k}}^{∗} (z_{1+n_{k}}) = 1 and

z_{1+n_{k}} ∈ P_{k}  and  z_{1+n_{k}}^{∗} ∈ Q_{k} .

GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 15

Find a natural number m_{k} larger than one so that

2^{−m_{k}/2} max ( ∥z_{1+n_{k}}∥ , ∥z_{1+n_{k}}^{∗} ∥ ) < min ( ε , δ_{k} )

and let

J_{k} := {1 + n_{k}, …, 2^{m_{k}} + n_{k}} and so J_{k}^{o} := {2 + n_{k}, …, 2^{m_{k}} + n_{k}} .

Let

P̃_{k} := P_{k} ∩ [z_{1+n_{k}}^{∗}]^{⊤}  and  Q̃_{k} := Q_{k} ∩ [z_{1+n_{k}}]^{⊥} .

The next step is to find a biorthogonal system {z_{n}, z_{n}^{∗}}_{n∈J_{k}^{o}} along with {i_{n}}_{n∈J_{k}^{o}} and {β_{n}}_{n∈J_{k}^{o}} satisfying

\{z_{n}, z_{n}^{∗}\} ∈ S \binom{}{P_{k}} × (2 + ε) B \binom{}{Q_{k}}  (1)

and

∥\frac{z_{n}^{∗}}{β_{n}} − w_{i_{n}}^{∗}∥ < δ_{k}  (2)

for each n ∈ J_{k}^{o}. Towards this, fix j ∈ J_{k}^{o} and assume that a biorthogonal system {z_{n}, z_{n}^{∗}}_{2+n_{k}⩽n<j} along with {i_{n}}_{2+n_{k}⩽n<j} and {β_{n}}_{2+n_{k}⩽n<j} have been constructed so that conditions (1) and (2) hold for 2+n_{k} ⩽ n < j. Let

𝔛_{j} := P̃_{k} ∩ [z_{n}^{∗}]_{2+n_{k}⩽n<j}^{⊤}  and  Y_{j} := Q̃_{k} ∩ [z_{n}]_{2+n_{k}⩽n<j}^{⊥} .

By Lemma 9, there is a weak^{∗}-closed finite co-dimensional subspace Ỹ_{j} of 𝔛^{∗} such that Ỹ_{j} is (2+ε/2)-normed by 𝔛_{j}. Find i_{j} > i_{j−1} and y_{j}^{∗} ∈ S (Y_{j} ∩ Ỹ_{j}) such that

∥y_{j}^{∗} − w_{i_{j}}^{∗} ∥ < δ_{k} .

Find z_{j} ∈ S(𝔛_{j}) such that

\frac{1}{2 + ε} ⩽ y_{j}^{∗} (z_{j}) := \frac{1}{β_{j}}

and normalize

z_{j}^{∗} := β_{j} y_{j}^{∗} .

This completes the inductive construction of {z_{n}, z_{n}^{∗}}_{n∈J_{k}^{o}} along with the sets {i_{n}}_{n∈J_{k}^{o}} and {β_{n}}_{n∈J_{k}^{o}}.

Now apply the Haar matrix to {z_{n}, z_{n}^{∗}}_{n∈J_{k}} to produce {x_{n}, x_{n}^{∗}}_{n∈J_{k}}. With help from the observations in Remark 10, note that {x_{n}, x_{n}^{∗}}_{n∈J_{k}} is biorthogonal and is in P_{k} × Q_{k}. Furthermore, for each n in J_{k},

∥x_{n}∥ ⩽ 2^{−m_{k}/2} ∥z_{1+n_{k}}∥ + (1 + \sqrt{2}) max_{j∈J_{k}^{o}} ∥z_{j}∥
⩽ ε + (1 + \sqrt{2})

16 DILWORTH, GIRARDI, JOHNSON

and

∥x_{n}^{∗}∥ ⩽ 2^{−m_{k}/2} ∥z_{1+n_{k}}^{∗} ∥ + (1 + \sqrt{2}) max_{j∈J_{k}^{o}} ∥z_{j}^{∗}∥
⩽ ε + (2 + ε) (1 + \sqrt{2})

and for each x^{∗∗} ∈ S (𝔛^{∗∗})

|x^{∗∗} (x_{n}^{∗})| ⩽ 2^{−m_{k}/2} ∥z_{1+n_{k}}^{∗} ∥ + (1 + \sqrt{2}) max_{j∈J_{k}^{o}} |x^{∗∗} (z_{j}^{∗})|

⩽ δ_{k} + (1 + \sqrt{2}) max_{j∈J_{k}^{o}} ( |x^{∗∗} (β_{j}w_{i_{j}}^{∗})| + |β_{j}δ_{k}| ) .

Thus

{x_{n}, x_{n}^{∗}}_{n∈J_{k}^{p} ∪J_{k}}

with {i_{n}}_{n∈J_{k}^{po}∪J_{k}^{o}} and {β_{n}}_{n∈J_{k}^{po}∪J_{k}^{o}} satisfy conditions (1) through (4). If k is odd, then

[a_{h}]_{h⩽h_{k}} ∈ [x_{n}, z_{1+n_{k}}]_{n∈J_{k}^{p}} ⊂ [x_{n}]_{n∈J_{k}^{p}∪J_{k}} ,

while if k is even, then

[b_{h}^{∗}]_{h⩽h_{k}} ∈ [x_{n}^{∗}, z_{1+n_{k}}^{∗}]_{n∈J_{k}^{p}} ⊂ [x_{n}^{∗}]_{n∈J_{k}^{p}∪J_{k}} .

Clearly the constructed system {x_{n}, x_{n}^{∗}}_{n=1}^{∞}, with the blocking {J_{k}}_{k=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.

Remark 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}) in 𝔛^{∗} 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; Z̃_{2} denotes Z_{2} equipped with the new ||| · |||-norm. Since Z̃_{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 Z̃_{2}. For example, a subset of Z_{2} is orthonormal if, when viewed as a subset of Z̃_{2}, it is orthonormal in Z̃_{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}. ■

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 taken 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 ∥_{ℓ_{1}⊕_{1}L_{1}} ⩽ ∥ T z ∥_{𝔛^{∗}} ⩽ (1 + ε) ∥ z ∥_{ℓ_{1}⊕_{1}L_{1}} .

Moreover, the image {T δ_{n}} of the unit vector basis of ℓ_{1} can be assumed to be weak^{∗}-null (since 𝔛 is separable, {T δ_{n}} has a weak^{∗}-convergent subsequence \{T δ_{k_{n}}\}, so just replace δ_{n} by \frac{1}{2}(δ_{k_{2n}} − δ_{k_{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

y_{n}^{∗} = T δ_{n} + βx_{n}^{∗}

and

Z_{1} := [ {y_{n}^{∗} : n ∈ ℕ} ∪ T L_{1} ] .

Note that for each n ∈ ℕ,

∥y_{n}^{∗}∥ ⩽ 1 + ε + β .  (3)

The operator S: ℓ_{1} ⊕_{1} L_{1} → Z_{1} defined by

S (∑ α_{n}δ_{n} ⊕_{1} f) = ∑ α_{n}y_{n}^{∗} + T f

illustrates that Z_{1} is isomorphic to ℓ_{1} ⊕_{1} L_{1}. Indeed, fix

∑ α_{n}δ_{n} ⊕_{1} f ∈ sp{δ_{n}}_{n∈ℕ} ⊕_{1} L_{1} .

Then

∥S (∑ α_{n}δ_{n} ⊕_{1} f)∥_{𝔛^{∗}} ⩽ (1 + ε + β) ∑ |α_{n}| + (1 + ε) ∥f∥_{L_{1}}
⩽ (1 + ε + β) ∥∑ α_{n}δ_{n} ⊕_{1} f∥_{ℓ_{1}⊕_{1}L_{1}} .

On the other hand,

∥S (∑ α_{n}δ_{n} ⊕_{1} f)∥_{𝔛^{∗}} = ∥T (∑ α_{n}δ_{n} ⊕_{1} f) + β ∑ α_{n}x_{n}^{∗}∥_{𝔛^{∗}}
⩾ ∥∑ α_{n}δ_{n} ⊕_{1} f∥_{ℓ_{1}⊕_{1}L_{1}} − β ∥∑ α_{n}δ_{n}∥_{ℓ_{1}}
⩾ (1 − β) ∥∑ α_{n}δ_{n} ⊕_{1} f∥_{ℓ_{1}⊕_{1}L_{1}} .

Thus Z_{1} is (\frac{1+β+ε}{1−β}) -isomorphic to ℓ_{1} ⊕_{1} L_{1}, which is 3-isomorphic to L_{1}.

To see that \{y_{n}^{∗}\}_{n∈ℕ} is (1 + \frac{1+ε}{β})-norming for 𝔛, fix x ∈ S(𝔛). Let δ > 0 be such that δ(1 + δ) < β and find n ∈ ℕ such that

|(T δ_{n})(x)| ⩽ δ  and  1 ⩽ (1 + δ)x_{n}^{∗}(x) ,  (4)

for then by (3) and (4)

\frac{y_{n}^{∗}(x)}{∥y_{n}^{∗}∥} ⩾ \frac{1}{1 + ε + β} (\frac{β}{1 + δ} − δ) .

Thus

sup_{n∈ℕ} \frac{y_{n}^{∗}(x)}{∥y_{n}^{∗}∥} ⩾ \frac{β}{β + 1 + ε} .

So, for β sufficiently close to one, {y_{n}^{∗}}_{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 Remark 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 + \sqrt{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 + \sqrt{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 Dvoretzky [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_{2}^{N} and a projection P from E + F onto F with ker P = E and ∥P ∥ < 1 + δ.

In fact, they showed that P can be taken 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.

GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 19

Fix a strictly decreasing sequence {η_{k}}_{k=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 blocking {J_{k}}_{k=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 + \sqrt{2}) + ε

(3) ∥x_{n}^{∗}∥ ⩽ 1 + ε

(4) ∥x_{n}^{∗} − u_{n}^{∗}∥ ⩽ η_{k} if n ∈ J_{k}

(5) [u_{n}^{∗}]_{n∈J_{k_{1}}} is orthogonal to [u_{n}^{∗}]_{n∈J_{k_{2}}} for k_{1} ≠ k_{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 weakly-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

δ := \frac{ε}{2 + \sqrt{2}} .

The construction will inductively produce blocks

{x_{n}, x_{n}^{∗}}_{n∈J_{k}}  and  {u_{n}^{∗}}_{n∈J_{k}} .

Fix k ⩾ 1. Assume that {J_{j}}_{0⩽j<k} along with {x_{n}, x_{n}^{∗}}_{n∈J_{k}^{p}} and {u_{n}^{∗}}_{n∈J_{k}^{p}} have been constructed to satisfy conditions (1) through (5). Now to construct J_{k} along with {x_{n}, x_{n}^{∗}}_{n∈J_{k}} and {u_{n}^{∗}}_{n∈J_{k}}.

The idea is to find a biorthogonal system {z_{n}, z_{n}^{∗}}_{n∈J_{k}} in P_{k} × Q_{k} by first finding {z_{1+n_{k}}, z_{1+n_{k}}^{∗}} that helps guarantee condition (6) if k is odd and condition (7) if k even; however, {z_{1+n_{k}}, z_{1+n_{k}}^{∗}} would not necessarily satisfy conditions (2) and (3) and z_{1+n_{k}}^{∗} may be far from Z_{2} and so J_{k}^{o} and

{z_{n}, z_{n}^{∗}}_{n∈J_{k}^{o}}

are then constructed and the Haar matrix is applied to {z_{n}, z_{n}^{∗}}_{n∈J_{k}} to produce {x_{n}, x_{n}^{∗}}_{n∈J_{k}} and {u_{n}^{∗}}_{n∈J_{k}} so that

{x_{n}, x_{n}^{∗}}_{n∈J_{k}^{p} ∪J_{k}}  and  {u_{n}^{∗}}_{n∈J_{k}^{p} ∪J_{k}}

satisfy conditions (1) through (5).

Find {z_{1+n_{k}}, z_{1+n_{k}}^{∗}} just as in the proof of Theorem 12: in the case that k is odd, be sure to choose y_{1+n_{k}}^{∗} in Z_{1}, which is possible since Z_{1} is total.

Find a natural number m_{k} larger than one so that

2^{−m_{k}/2} max ( ∥z_{1+n_{k}}∥ , ∥z_{1+n_{k}}^{∗} ∥ ) < min ( δ , η_{k} ) .

Let

E_{k} := [{x_{n}^{∗}}_{n∈J_{k}^{p}} ∪ {z_{1+n_{k}}^{∗}}]
H_{k} := ( Z_{2} ⊖ [u_{n}^{∗}]_{n∈J_{k}^{p}} ) ∩ [x_{n}]_{n∈J_{k}^{p}}^{⊥} ∩ [z_{1+n_{k}}]^{⊥}
Y_{k} := [Z_{2} ∪ E_{k}]
N_{k} := 2^{m_{k}} − 1 = |J_{k}^{o}| .

Use Fact 16 to find a subspace F_{k} of H_{k}, a projection P_{k} with kernel E_{k}, and a norm one isomorphism T_{k} so that

E_{k} + F_{k} ↠^{P_{k}} F_{k} ⟶^{T_{k}} ℓ_{2}^{N_{k}}

and

max ( ∥T_{k}^{−1}∥ , ∥P_{k}∥^{2} ) < 1 + δ .

Let {e_{n}}_{n∈J_{k}^{o}} be an orthonormal basis for ℓ_{2}^{N_{k}}. For each n ∈ J_{k}^{o}, let

z_{n}^{∗} = T_{k}^{−1}e_{n}

and, using Local Reflexivity, find z_{n} ∈ 𝔛 that agrees, on E_{k} and F_{k}, with a norm-preserving Hahn-Banach extension of P_{k}^{∗}T_{k}^{∗}e_{n} ∈ (E_{k} + F_{k})^{∗} to 𝔛^{∗} and satisfies

∥z_{n}∥ < \sqrt{1 + δ} ∥P_{k}^{∗}T_{k}^{∗}e_{n}∥ .

Then {z_{n}, z_{n}^{∗}}_{n∈J_{k}^{o}} is a biorthogonal system in E_{k}^{⊤} × F_{k} and

max { ∥z_{n}∥ : n ∈ J_{k}^{o} } < 1 + δ .

Now apply the Haar matrix to {z_{n}, z_{n}^{∗}}_{n∈J_{k}} to produce {x_{n}, x_{n}^{∗}}_{n∈J_{k}} and let

u_{n}^{∗} := ∑_{j∈J_{k}^{o}} a_{nj}^{m_{k}}z_{j}^{∗}

for each n in J_{k}.

With help from the observations in Remark 10, note that for each n in J_{k}

∥x_{n}∥ ⩽ 2^{−m_{k}/2} ∥z_{1+n_{k}}∥ + (1 + \sqrt{2}) max_{j∈J_{k}^{o}} ∥z_{j}∥
⩽ δ + (1 + \sqrt{2}) (1 + δ)
⩽ (1 + \sqrt{2}) + ε

and

∥x_{n}^{∗}∥ ⩽ 2^{−m_{k}/2} ∥z_{1+n_{k}}^{∗} ∥ + ∥ ∑_{j∈J_{k}^{o}} a_{nj}^{m_{k}}z_{j}^{∗}∥

⩽ δ + (1 + δ) ∥ ∑_{j∈J_{k}^{o}} a_{nj}^{m_{k}}e_{j}∥

⩽ 1 + 2 δ ⩽ 1 + ε .

and since x_{n}^{∗} − u_{n}^{∗} = a_{n1}^{m_{k}} z_{n_{k}+1}^{∗},

∥x_{n}^{∗} − u_{n}^{∗}∥ = 2^{−m_{k}/2} ∥z_{n_{k}+1}^{∗}∥ ⩽ η_{k} .

Thus,

{x_{n}, x_{n}^{∗}}_{n∈J_{k}^{p} ∪J_{k}}  and  {u_{n}^{∗}}_{n∈J_{k}^{p} ∪J_{k}}

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 blocking {J_{k}} 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

[figure: ℓ_{1}; 𝔛^{∗}; L_{1}; A; j; B]

where j is an isomorphic embedding with range Z_{1} and A and B are given by

A(δ_{n}) = j_{o}^{−1}x_{n}^{∗} := r̃_{n} and B(δ_{n}) = x_{n}^{∗} .

Since A^{∗} and B^{∗} are of the form

A^{∗}(f) = (f \binom{}{r_{n}})_{n}  and  B^{∗}(x^{∗∗}) = (x^{∗∗} (x_{n}^{∗}))_{n} ,

their ranges are contained in c_{0}; let A_{o}^{∗} and B_{o}^{∗} be the corresponding maps with their ranges restricted to c_{0}. Thus the following diagram commutes.

[figure: 𝔛; 𝔛^{∗∗}; c_{0}; L_{∞}; j^{∗}; A_{o}^{∗}; B_{o}^{∗}; δ; S]

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 weaker 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 weakly-null sequence {w_{n}^{∗}} 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}}_{n⩾1} in S(𝔛)

(b) a sequence {x_{n}^{∗}}_{n⩾1} in S(𝔛^{∗})

(c) finite sets {F_{n}}_{n⩾0} in S(𝔛^{∗}) with F_{0} := ∅

(d) an increasing sequence {k_{n}}_{n=1}^{∞} of integers that satisfy

(4) x_{n} ∈ F_{n−1}^{⊤} ∩ {x_{i}^{∗}}_{i<n}^{⊤} := 𝔛_{n}

(5) x_{n}^{∗} ∈ {d_{i}}_{i<n}^{⊥} ∩ {x_{i}}_{i<n}^{⊥} := Y_{n}

GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 23

(6) \frac{1}{2+ε} ⩽ x_{n}^{∗}(x_{n})

(7) ∥x_{n}^{∗} − w_{k_{n}}^{∗}∥ ⩽ ε_{n}

(8) if x ∈ [x_{j}]_{j⩽n}, then there is f ∈ F_{n} with ∥x∥ ⩽ τ_{n}f(x). For then just take 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

d_{i} = (d_{i} − ∑_{n=1}^{i} x_{n}^{∗}(d_{i})x_{n}) + ∑_{n=1}^{i} x_{n}^{∗}(d_{i})x_{n} .

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 + \frac{ε}{2})-normed by 𝔛_{n}. Find x_{n}^{∗} ∈ S (Y ∩ Y_{n}) along with k_{n} > k_{n−1} such that (7) holds. Since Y is (2 + \frac{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. Take 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)}) y_{n} .

Clearly {w_{n}, x_{n}^{∗}} is a [(1 + ε)(2 + ε)]-bounded wc_{0}^{∗}-biorthogonal system.

Fix n_{0} ∈ ℕ and consider x^{∗} ∈ [w_{p(n_{0},i)}]_{i}^{⊥}. For each m ∈ ℕ,

x^{∗} (∑_{i=1}^{m} a_{p(n_{0},i)}x_{p(n_{0},i)}) = εx^{∗}(y_{n_{0}}) ∑_{i=1}^{m} |a_{p(n_{0},i)}| .

Combined with (ii) and (iii), this gives that x^{∗} ∈ [y_{n_{0}}]^{⊥}, which in turn implies that x^{∗} ∈ [x_{p(n_{0},i)}]_{i}^{⊥}. Thus

[ {x_{p(n_{0},i)}}_{i} ∪ {y_{n_{0}}} ] ⊂ [w_{p(n_{0},i)}]_{i} .

Combined with (3), it follows that {w_{n}}_{n∈ℕ} is fundamental. ■

5. BOUNDED FUNDAMENTAL BIORTHOGONAL SYSTEMS

The knowledgeable reader notices that, in our proof of Theorem 15, a combination of the [OP]–method (which produces [(1 + \sqrt{2})^{2} + ε]-bounded systems) and the [P]–method (which produces (1 + ε)-bounded systems) is used to produce a (1 + \sqrt{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 + \sqrt{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 + \sqrt{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 + \sqrt{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^{∗}

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 weak^{∗}-null (similar to before, just replace \{e_{n}^{∗}\} with a weak^{∗}-convergent subsequence \{\frac{1}{2}(e_{k_{2n}}^{∗} −e_{k_{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

y_{n}^{∗} = e_{n}^{∗} + βx_{n}^{∗}

and

Z_{1} := [ {y_{n}^{∗} : n ∈ ℕ} ∪ ker P ] .

Note that for each n ∈ ℕ,

∥y_{n}^{∗}∥ ⩽ 1 + β .  (5)

Each element in Z has a unique expression as

z = ∑ α_{n}e_{n}^{∗} + f

26 DILWORTH, GIRARDI, JOHNSON

where f ∈ ker P; the operator S: Z → Z_{1} defined by

S (∑ α_{n}e_{n}^{∗} + f) = ∑ α_{n}y_{n}^{∗} + f

illustrates that Z_{1} is isomorphic to Z. Indeed, fix

∑ α_{n}e_{n}^{∗} + f ∈ sp{e_{n}^{∗}}_{n∈ℕ} + ker P .

Then

∥z − Sz∥ = ∥∑ α_{n}(e_{n}^{∗} − y_{n}^{∗})∥
⩽ β ∑ |α_{n}| = β ∥P (z)∥
⩽ β ∥z∥ .

Thus

(1 − β) ∥z∥ ⩽ ∥Sz∥ ⩽ (1 + β) ∥z∥

for each z ∈ Z; thus, Z_{1} is (\frac{1+β}{1−β}) -isomorphic to Z.

To see that \{y_{n}^{∗}\}_{n∈ℕ} is (1 + \frac{1}{β} )-norming for 𝔛, fix x ∈ S(𝔛). Let δ > 0 be such that δ(1 + δ) < β and find n ∈ ℕ such that

|e_{n}^{∗}(x)| ⩽ δ  and  1 ⩽ (1 + δ)x_{n}^{∗}(x) ,  (6)

for then by (5) and (6)

\frac{y_{n}^{∗}(x)}{∥y_{n}^{∗}∥} ⩾ \frac{1}{1 + β} (\frac{β}{1 + δ} − δ) .

Thus

sup_{n∈ℕ} \frac{y_{n}^{∗}(x)}{∥y_{n}^{∗}∥} ⩾ \frac{β}{β + 1} .

So \{y_{n}^{∗}\}_{n∈ℕ} is (1 + \frac{1}{β})-norming for 𝔛. ■

Recall that the modulus of convexity δ_{W}: [0, 2] → [0, 1] of a Banach space W is

δ_{W}(ε) := inf \{1 − ∥\frac{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. 53–55]).

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.

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

∑_{j=1}^{∞} |α_{j}| = 1 and  ∥∑_{j=1}^{∞} α_{j}y_{j}∥ > 1 − ε ,  (7)

then there is a finite subset F of ℕ so that

∑_{j∈F} |α_{j}| > 1 − \frac{ε}{2b − ε}  (8)

and for each j ∈ F,

∥(sign α_{j}) y_{j} − ∑_{j=1}^{∞} α_{j}y_{j}∥ < δ_{W}^{−1} (b)  (9)

and α_{j} ≠ 0 .

Proof. Let {α_{j}}_{j=1}^{∞} ⊂ ℝ and {y_{j}}_{j=1}^{∞} ⊂ B(W) satisfy (7). Set

x_{0} := ∑_{j=1}^{∞} α_{j}y_{j} .

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 < ∑_{j∉F} α_{j}x_{0}^{∗}(y_{j}) + ∑_{j∈F} α_{j}x_{0}^{∗}(y_{j})

⩽ (1 + ε − 2b) (1 − ∑_{j∈F} α_{j}) + (∑_{j∈F} α_{j})

= (1 + ε − 2b) + (2b − ε) (∑_{j∈F} α_{j}) ,

condition (8) holds. For each j ∈ F,

\frac{1}{2} ∥y_{j} + x_{0}∥ > \frac{1}{2} ( (1 + ε − 2b) + (1 − ε) ) = 1 − b

and so

∥y_{j} − x_{0}∥ < δ_{W}^{−1} (b)

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^{∗} (K B (W^{∗}))) < 2η  (10)

for some constants K ≥ 1 and η > 0. If

1 − \frac{1 − 2η}{K} < \frac{2}{3} δ_{W} (\frac{1 − 4η}{2K} )  (11)

then {x_{n}}_{n=1}^{∞} is equivalent to the standard unit vector basis of ℓ_{1}. More specifically if constants a and b satisfy

\frac{3}{2} (1 − \frac{1 − 2η}{K} ) ⩽ a < b ⩽ δ_{W} (\frac{1 − 4η}{2K} )  (12)

then

\frac{3 (b − a)}{3b − a} ∑_{n=1}^{∞} |β_{n}| ⩽ ∥∑_{n=1}^{∞} β_{n}x_{n}∥ ⩽ ∑_{n=1}^{∞} |β_{n}|  (13)

for each {β_{n}}_{n=1}^{∞} in ℓ_{1}.

Note that if K = 1 and η = 0 then (11) becomes

0 < \frac{2}{3} δ_{W} (\frac{1}{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 {x̃_{n}}_{n=1}^{∞} of {x_{n}}_{n=1}^{∞} that are disjointly supported on the standard unit vector basis of ℓ_{1}. For then, {x̃_{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 {w_{n}^{∗}}_{n=1}^{∞} in K B (W^{∗}) so that

∥x_{n}^{∗} − Q^{∗}w_{n}^{∗}∥ < 2η .  (14)

Thus

∥Qx_{n}∥ ⩾ |⟨Qx_{n}, \frac{w_{n}^{∗}}{∥w_{n}^{∗}∥}⟩|

⩾ \frac{1}{∥w_{n}^{∗}∥} |⟨x_{n}, x_{n}^{∗}⟩ − ⟨x_{n}, x_{n}^{∗} − Q^{∗}w_{n}^{∗}⟩|

> \frac{1 − 2η}{K}

GEOMETRY OF BANACH SPACES AND BIORTHOGONAL SYSTEMS 29

and so by the first inequality in (12)

∥Qx_{n}∥ > 1 − 2a/3 .

Write

x_{n} = ∑_{j=1}^{∞} α_{j}^{n}δ_{j}

where ∑_{j=1}^{∞} |α_{j}^{n}| = 1 and let ε_{j}^{n} = sign α_{j}^{n}; thus,

Qx_{n} = ∑_{j=1}^{∞} α_{j}^{n} Qδ_{j} .

So by Lemma 21 there is a sequence {F_{n}}_{n=1}^{∞} of finite subsets in ℕ so that

∑_{j∈F_{n}} |α_{j}^{n}| > 1 − \frac{a}{3b − a}

and for each j ∈ F_{n}

∥ε_{j}^{n}Qδ_{j} − Qx_{n}∥ < δ_{W}^{−1} (b)

and α_{j}^{n} ≠ 0. Let

x̃_{n} = ∑_{j∈F_{n}} α_{j}^{n} δ_{j} .

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. Pick τ ∈ {−1, 1} so that τε_{j_{0}}^{m} = ε_{j_{0}}^{n}. Then

∥Qx_{n} − τQx_{m}∥ ⩽ ∥Qx_{n} − ε_{j_{0}}^{n}Qδ_{j_{0}} ∥ + ∥τε_{j_{0}}^{m}Qδ_{j_{0}} − τQx_{m}∥
< 2δ_{W}^{−1} (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}) , \frac{w_{n}^{∗}}{∥w_{n}^{∗}∥}⟩|

⩾ \frac{1}{∥w_{n}^{∗}∥} |⟨x_{n} ± x_{m}, x_{n}^{∗}⟩ − ⟨x_{n} ± x_{m}, x_{n}^{∗} − Q^{∗}w_{n}^{∗}⟩|

> \frac{1 − 4η}{K}

⩾ 2δ_{W}^{−1} (b) .

A contradiction, thus the finite subsets {F_{n}}_{n=1}^{∞} of ℕ are indeed disjoint.

For each {β_{n}}_{n=1}^{∞} in ℓ_{1},

∑_{n=1}^{∞} |β_{n}| ⩾ ∥∑_{n=1}^{∞} β_{n}x_{n}∥

⩾ ∥∑_{n=1}^{∞} β_{n}x̃_{n}∥ − ∑_{n=1}^{∞} |β_{n}| ∥x̃_{n} − x_{n}∥

⩾ (1 − \frac{a}{3b − a}) ∑_{n=1}^{∞} |β_{n}| − (\frac{a}{3b − a}) ∑_{n=1}^{∞} |β_{n}|

= (1 − \frac{2a}{3b − a}) ∑_{n=1}^{∞} |β_{n}|

and so (13) holds. ■

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

(∑_{i=1}^{n} ∥w_{i}^{∗}∥^{q})^{1/q} ⩽ C_{q} (W^{∗}) [ avg_{θ_{i}=±1} ∥∑_{i=1}^{n} θ_{i}w_{i}^{∗}∥^{2} ]^{1/2} (15)

for each finite sequence {w_{i}^{∗}}_{i=1}^{n} in W^{∗}.

Theorem 23. Let a uniformly convex Banach space W and ε_{0} > 0 satisfy

1 − \frac{1}{1 + ε_{0}} = \frac{2}{3} δ_{W} (\frac{1}{2(1 + ε_{0})}) .  (16)

Let C_{q} (W^{∗}) be as in (15) and 0 < ε_{1} < ε_{0}. Then there exists a constant

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.

Notation. Consider the functions l, u: [0, 1/4]×[0, ∞)×[0, ∞) → [0, 1] given by

\begin{aligned}l(η_{1}, η_{2}, ε) & = 1 − \frac{1 − 2η_{1}}{(1 + η_{2})(1 + ε)} \\ u(η_{1}, η_{2}, ε) & = \frac{2}{3} δ_{W} (\frac{1 − 4η_{1}}{2(1 + η_{2})(1 + ε)}) . \end{aligned}  (18)

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

l(0, 0, ε_{0}) = u(0, 0, ε_{0})

and

l(0, 0, ·) : [0, ∞) ⟶^{onto} [0, 1)
u(0, 0, ·) : [0, ∞) ⟶^{onto} (0, \frac{2}{3} δ_{W} (\frac{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 + \frac{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 + \frac{2a}{3(b − a)})]^{q} < N ∈ ℕ

and 0 < η ⩽ \frac{1}{N} then (19) holds.

Let conditions (23a) through (23d) hold. Assume that ε ⩽ ε_{1}.

32 DILWORTH, GIRARDI, JOHNSON

By (23c), without loss of generality, for each w^{∗} ∈ W^{∗},

\frac{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 {w_{n}^{∗}}_{n=1}^{∞} ⊂ KB(W^{∗}) such that

∥x_{n}^{∗} − Q^{∗}w_{n}^{∗}∥ < 2η
∥x_{n}^{∗}∥ = ∥Q^{∗}w_{n}^{∗}∥ .

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

\frac{3 (b − a)}{3b − a} ∥∑_{n=1}^{∞} β_{n}x_{n}^{∗}∥ ⩽ sup_{n} |β_{n}| ⩽ ∥∑_{n=1}^{∞} β_{n}x_{n}^{∗}∥

for each {β_{n}}_{n=1}^{∞} in c_{0}.

For each finite subset F of ℕ

[∑_{n∈F} ∥w_{n}^{∗}∥^{q}]^{\frac{1}{q}} ⩽ C [ avg_{θ_{i}=±1} ∥∑_{i=1}^{n} θ_{i}w_{i}^{∗}∥^{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

∥∑_{n∈F} θ_{n}w_{n}^{∗}∥ ⩽ (1 + η) ∥∑_{n∈F} θ_{n}Q^{∗}w_{n}^{∗}∥

⩽ (1 + η) [ ∥∑_{n∈F} θ_{n}x_{n}^{∗}∥ + ∥∑_{n∈F} θ_{n} (x_{n}^{∗} − Q^{∗}w_{n}^{∗})∥ ]

⩽ (1 + η) [\frac{3b − a}{3(b − a)} + 2 |F | η] .

The left-hand side of (21) mimics l_{q}-growth since

1 ⩽ ∥x_{n}^{∗}∥ = ∥Q^{∗}w_{n}^{∗}∥ ⩽ ∥w_{n}^{∗}∥

for each n ∈ ℕ. Thus

|F |^{\frac{1}{q}} ⩽ C(1 + η) [\frac{3b − a}{3(b − a)} + 2 |F | η] .

This contradicts (19). Thus ε_{1} < ε. ■

Corollary 24. Let

0 < ε_{1} < −2 + \frac{\sqrt{147}}{6}

and, following the notation in (17),

η = η (1, δ_{ℓ_{2}}, 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}. Thus C_{2}(W^{∗}) = 1. It is straight forward to verify that

ε_{0} := −2 + \frac{\sqrt{147}}{6}

satisfies condition (16) of Theorem 23. Note that ε_{0} ≈ 0.0207.

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 weak^{∗}-to-weak^{∗} continuous. Similarly, by reflexivity, there is S ∈ L(^{∗}Z_{2}, ℓ_{2}) such that S^{∗} ∈ L(ℓ_{2}, Z_{2}) is a (1 + η)–isomorphism. Let Q = S ∘ T. Thus

Q: ℓ_{1} →^{T} ^{∗}Z_{2} →^{S} ℓ_{2}  and  Q^{∗}: ℓ_{2} →^{S^{∗}} Z_{2} →^{T^{∗}} ℓ_{∞}

and Q^{∗} is a (1 + η)–isomorphic embedding.

Thus condition (22) follows from Theorem 23. ■

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Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.

E-mail address: dilworth@math.sc.edu and girardi@math.sc.edu

Department of Mathematics, Texas A&M University, College Station, TX 77843, U.S.A.

E-mail address: johnson@math.tamu.edu