Espaces de Banach stables

We define the notion of “stable Banach space” by a simple condition on the norm. We prove that ifE is a stable Banach space, then every subspace ofL ^p(E) (1≦p<∞) is stable. Our main result asserts that every infinite dimensional stable Banach space containsl ^p, for somep, 1≦p<∞. This is a generalization of a theorem due to D. Aldous: every infinite dimensional subspace ofL ¹ containsl ^p, for somep in the interval [1, 2].     

On countable strict inductive limits

In this paper we prove that if E is the strict inductive limit of a sequence of Mackey spaces {E_n} such that for every positive integer n, the topological dual space of E_n, E′_n, provided with the Mackey topology μ(E′_n,E_n), is ultrabornological, then the topological dual space E′ of E, provided with the Mackey topology μ(E′,E), is ultrabornological. We also prove that if E is a strict (LF)-space and G a closed subspace of E′ [μ(E′,E)] such that E′[μ(E′,E)] /G is sequentially complete, then E′[μ(E′,E)]/G is complete.     

A class of bornological barrelled spaces which are not ultrabornological

N. Bourbaki, [1, p. 35], notices that it is not known if every bornological barrelled space is ultrabornological. In this paper we prove that if E is the topological product of an infinite family of bornological barrelled spaces, of non-zero dimension, there exists an infinite number of bornological barrelled subspaces ofE, which are not ultrabornological. We also give some examples of barrelled normable spaces which are not ultrabornological.Supported in part by the Patronato para el Fomento de la Investigación en la Universidad.     

A non-reflexive Grothendieck space that does not containl ∞

A compact spaceS is constructed such that, in the dual Banach spaceC(S)^*, every weak^* convergent sequence is weakly convergent, whileC(S) does not have a subspace isomorphic tol _∞. The construction introduces a weak version of completeness for Boolean algebras, here called the Subsequential Completeness Property. A related construction leads to a counterexample to a conjecture about holomorphic functions on Banach spaces. A compact spaceT is constructed such thatC(T) does not containl _∞ but does have a “bounding” subset that is not relatively compact. The first of the examples was presented at the International Conference on Banach spaces, Kent, Ohio, 1979.     

Asplund operators and holomorphic maps

LetE andF be locally convex topological vector spaces. A holomorphic mapf: E→F is defined to be an Asplund map if it takes the separable subsets of a neighbourhood of eacha∈E into absolutely convex weakly metrisable subsets ofF; a Banach space is an Asplund space if and only if its identity map has this property. We show that a continuous linear map from a quasinormable locally convex spaceE into a Banach spaceF is an Asplund map if and only if it factors through an Asplund space. IfE andF are both Banach spaces, then a holomorphic mapf: E→F is an Asplund map if and only if its derivative maps factor through Asplund spaces for eacha∈E. This is true if and only if such a factorisation holds ata=0. Part of this research was done during a visit to the University of Namibia, whose financial support is gratefully acknowledged This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990     

On smooth, nonlinear surjections of banach spaces

It is shown that (1) every infinite-dimensional Banach space admits aC ¹ Lipschitz map onto any separable Banach space, and (2) if the dual of a separable Banach spaceX contains a normalized, weakly null Banach-Saks sequence, thenX admits aC ^∞ map onto any separable Banach space. Subsequently, we generalize these results to mappings onto larger target spaces. Supported by an NSF Postdoctoral Fellowship in Mathematics.     

On weakly null FDD'S in Banach spaces

In this paper we show that every sequence (F _n ) of finite dimensional subspaces of a real or complex Banach space with increasing dimensions can be “refined” to yield an F.D.D. (G _n ), still having increasing dimensions, so that either every bounded sequence (x _n ), withx _n ∈G _n forn∈ℕ, is weakly null, or every normalized sequence (x _n ),withx _n ∈G _n forn∈ℕ, is equivalent to the unit vector basis of ℓ₁. Crucial to the proof are two stabilization results concerning Lipschitz functions on finite dimensional normed spaces. These results also lead to other applications. We show, for example, that every infinite dimensional Banach spaceX contains an F.D.D. (F _n ),with lim _n→∞dim(F _n )=∞, so that all normalized sequences (x _n ),withx _n ∈F _n ,n∈∕, have the same spreading model overX. This spreading model must necessarily be 1-unconditional overX. Research partially supported by NSF DMS-8903197, DMS-9208482, and TARP 235. Research partially supported by NSF.     

Embedding ofl ∞ k in finite dimensional Banach spaces

Letx ₁,x ₂, ...,x _n ben unit vectors in a normed spaceX and defineM _n =Ave{‖Σ _i=1 ⁿ ε₁ x _i ‖:ε₁=±1}. We prove that there exists a setA⊂{1, ...,n} of cardinality such that {x _i } _i∈A is 16M _n -isomorphic to the natural basis ofl _∞ ^k . This result implies a significant improvement of the known results concerning embedding ofl _∞ ^k in finite dimensional Banach spaces. We also prove that for every ∈>0 there exists a constantC(∈) such that every normed spaceX _n of dimensionn either contains a (1+∈)-isomorphic copy ofl ₂ ^m for somem satisfying ln lnm≧1/2 ln lnn or contains a (1+∈)-isomorphic copy ofl _∞ ^k for somek satisfying ln lnk>1/2 ln lnn−C(∈). These results follow from some combinatorial properties of vectors with ±1 entries. The contribution of the first author to this paper forms part of his Ph.D. Thesis written under the supervision of Prof. M. A. Perles from the Hebrew University.     

The classification of injective Banach lattices

LetE be a 1-injective Banach lattice,X any Banach space andT: E ← X a norm bounded linear operator. Then eitherT is an isomorphism on some copy ofl _∞ inE or for all σ > 0 there is φ ∈E ₊ ^′ such that ‖Tu‖≦φ (|u|)+σ ‖u‖ for allu ∈E. We deduce the theorem that: A norm order continuous injective Banach lattice is order isomorphic to an (AL)-space.     

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach spaceℓ _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

A Schur test for weighted mixed-norm spaces

Summary We prove a Schur test for mixed-norm spaces L^p,q, 1 < p,q < ∞. Also we prove another version of the Schur test for discrete weighted mixed-norm spaces l^p,q _w, 1 < p,q < ∞, and wis a weight. We show that if w ₁, and w ₂are two weight functions on the index sets Jx Iand K x Lrespectively, and A =(a _{ji, kl} ) _{j∈J, i∈I, k∈K, l∈L} is an infinite matrix, then under certain conditions, Ais a bounded operator from l^p,q _w1, 1 < p,q < ∞ to l^p,q _w2. This will be a key result in proving boundedness of important operators in our work in time-frequency analysis.</o:p>     

An extension of a result of ribe

It is proved that for every 1≦p<∞, 1≦q<∞ and for every sequence {p _n}, 1≦p _n<∞,p _n→p, the spaceX=(Σ⊕l _{p n}) _q (resp.U=(Σ⊕L _{p n}(0, 1)) _q ) is uniformly homeomorphic toX⊕l _p (resp.U⊕L _p(0, 1)). This extends Ribe’s result from the casep=1 to generalp<∞ and thus provides examples of uniformly convex, uniformly homeomorphic Banach spaces which are not Lipschitz equivalent.     

A class of Banach sequence spaces analogous to the space of Popov

Hagler and the first named author introduced a class of hereditarily l ₁ Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily l _p Banach spaces for 1 ⩽ p < ∞. Here we use these spaces to introduce a new class of hereditarily l _p (c ₀) Banach spaces analogous of the space of Popov. In particular, for p = 1 the spaces are further examples of hereditarily l ₁ Banach spaces failing the Schur property.     

Sur les espaces de Banach contenantl 1(τ)

Letτ be a cardinal with cf(τ)>ℵ₀. Then a Banach spaceE contains a subspace isomorphic tol ^l(τ) if and only if [0,1] ^r is a continuous image of the unit ballE′₁ ofE′, provided with the w*-topology. It follows that, for each cardinalκ, ifE′₁ contains a copy ofβκ, thenE has a quotient isomorphic tol ^∞(κ). In this situation we show thatE has even a quotientisometric tol ^∞(κ).      

Structure of spaces ofC ∞-functions on nuclear spaces

LetE be a real nuclear locally convex space; we prove that the space ℰ_ub(E), of allC ^∞-functions of uniform bounded type onE, coincides with the inductive limit of the spaces ℰ_Nbc(E _v) (introduced by Nachbin-Dineen), whenV ranges over a basis of convex balanced 0-neighbourhoods inE. LetE be a real nuclear bornological vector space; we prove that the space ℰ(E) of allC ^∞-functions onE coincides with the projective limit of the spaces ℰ_Nbc(E _B), whenB is a closed convex balanced bounded subset ofE. As a consequence we obtain some density results and a version of the Paley-Wiener-Schwartz theorem. Research done during the stay of this author at the University of Bordeaux (France) in the academic year 1980–1981.     

Compactness in Vector-valued Banach Function Spaces

We give a new proof of a recent characterization by Diaz and Mayoral of compactness in the Lebesgue-Bochner spaces L_X^p, where X is a Banach space and 1≤ p<∞, and extend the result to vector-valued Banach function spaces E_X, where E is a Banach function space with order continuous norm. The author is supported by the ‘VIDI subsidie’ 639.032.201 in the ‘Vernieuwingsimpuls’ programme of the Netherlands Organization for Scientific Research (NWO) and by the Research Training Network HPRN-CT-2002-00281.     

Some new generalized difference sequence spaces defined by a sequence of moduli

The idea of difference sequence sets X( ) = {x = (x k ) : x ∈ X} with X = l ∞ , c and c 0 was introduced by Kizmaz [12]. In this paper, using a sequence of moduli we define some generalized difference sequence spaces and give some inclusion relations.     

New examples of <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript>-saturated Banach spaces

For every p ∈ (1, ∞), an isomorphically polyhedral Banach space E _p is constructed which has an unconditional basis and does not embed isomorphically into a C(K) space for any countable and compact metric space K. Moreover, E _p admits a quotient isomorphic to ℓ _p .     

Linear-topological classification of separableL p-spaces associated with von Neumann algebras of type I

We classify, up to a linear-topological isomorphism, all separableL _p-spaces, 1≤p<∞, associated with von Neumann algebras of type I. In particular, anyL _p-space associated with an infinite-dimensional atomic von Neumann algebra is isomorphic tol _p, or toC _p, or to . Further, anyL _p-space,p∈[1,∞),p∈2 associated with an infinite-dimensional von Neumann algebraM of type I is isomorphic to one of the following nine Banach spaces: l_p, L_p, S_P, C_p, S_p ⊕ L_p, L_p(S_p), C_p ⊕ L_p, L_p(C_p), C_p ⊕ L_p(S_p). In the casep=1 all the spaces in this list are pairwise non-isomorphic. Research supported by the Australian Research Council.     

Large subspaces ofl ∞ n and estimates of the gordon-lewis constantand estimates of the gordon-lewis constant

Lower bounds are obtained for thegl constants and hence also for the unconditional basis constants of subspaces of finite dimensional Banach spaces. Sharp results are obtained for subspaces ofl _∞ ⁿ , while in the general case thegl constants of “random large” subspaces are related to the distance of “random large” subspaces to Euclidean spaces. In addition, a new isometric characterization ofl _∞ ⁿ is given, some new information is obtained concerningp-absolutely summing operators, and it is proved that every Banach space of dimensionn contains a subspace whose projection constant is of ordern ^1/2. The research for this paper was begun while both authors were guests of the Mittag-Leffler Institute. Supported in part by NSF-MCS 79-03042.