A characterization of conjugate WCG banach spaces

It is proved that a WCG Banach space X is isomorphic to a conjugate Banach space if and only if there exists a closed subspace V of its conjugate space X' with positive characteristic such that X possesses the following summability property with respect to V: For every bounded sequence in X there exists a regular essentially positive summability method A such that the A-means of the sequence are σ(X,V)-convergent in X. This extends a well-known theorem of Nishiura-Waterman [8] and yields an analogous characterization of quasi-reflexive spaces. Conjugate spaces of smooth Banach spaces can also be characterized by the above summability condition.     

Universal uniform Eberlein compact spaces

A universal space is one that continuously maps onto all others of its own kind and weight. We investigate when a universal Uniform Eberlein compact space exists for weight . If , then they exist whereas otherwise, in many cases including , it is consistent that they do not exist. This answers (for many and consistently for all ) a question of Benyamini, Rudin and Wage of 1977.

     

On scattered Eberlein compact spaces

We prove that scattered Eberlein compacta of Cantor-Bendixson height at most ω + 1 are Uniform Eberlein compact spaces (ω + 1 is optimal for this result). For a set X and n ∈ ω, by σ _n (2^X) we denote the subspace of the product 2^X consisting of all characteristic functions of sets of cardinality ≥ n. We give an example of an Eberlein compactum K of weight ω _ω and of Cantor-Bendixson height 3 which cannot be embedded into any σ _n (2^X). Research of the first author supported by NSERC of Canada. Murray Bell died on December 9, 2001. Research of the second author supported by KBN grants 2 P03A 011 15 and 2 P03A 004 23. The main part of this research was done while the second author was visiting the University of Manitoba in 2000. He expresses his gratitude to the Department of Mathematics of U.M. for its hospitality.     

Linearly ordered Eberlein compact spaces

In this note we prove that every Eberlein compact linearly ordered space is metrizable. (By an Eberlein compact space we mean a topological space which can be embedded as a compact subset of a Banach space with the weak topology.)     

Embedding unconditional stable banach spaces into symmetric stable banach spaces

In this paper we prove the following result which solves a question raised by A. Pelczynski: “Every stable Banach space with an unconditional basis is isomorphic to a complemented subspace of some stable Banach space with a symmetric basis.” Moreover, we show that all the interpolation spacesl ^p ,l ^q _θ,X,1 1≦p, q<∞ andX stable, are stable.     

Examples concerning heredity problems of WCG Banach spaces

We present two examples of WCG spaces that are not hereditarily WCG. The first is a space with an unconditional basis, and the second is a space such that is WCG and does not contain . The non-WCG subspace of has the additional property that is not WCG and is reflexive.

     

Semisummands and semiideals in banach spaces

Let |·| be a fixed absolute norm onR ². We introduce semi-|·|-summands (resp. |·|-summands) as a natural extension of semi-L-summands (resp.L-summands). We prove that the following statements are equivalent. (i) Every semi-|·|-summand is a |·|-summand, (ii) (1, 0) is not a vertex of the closed unit ball ofR ² with the norm |·|. In particular semi-L ^p-summands areL ^p-summands whenever 1<p≦∞. The concept of semi-|·|-ideal (resp. |·|-ideal) is introduced in order to extend the one of semi-M-ideal (resp.M-ideal). The following statements are shown to be equivalent. (i) Every semi-|·|-ideal is a |·|-ideal, (ii) every |·|-ideal is a |·|-summand, (iii) (0, 1) is an extreme point of the closed unit ball ofR ² with the norm |·|. From semi-|·|-ideals we define semi-|·|-idealoids in the same way as semi-|·|-ideals arise from semi-|·|-summands. Proper semi-|·|-idealoids are those which are neither semi-|·|-summands nor semi-|·|-ideals. We prove that there is a proper semi-|·|-idealoid if and only if (1, 0) is a vertex and (0, 1) is not an extreme point of the closed unit ball ofR ² with the norm |·|. So there are no proper semi-L ^p-idealoids. The paper concludes by showing thatw*-closed semi-|·|-idealoids in a dual Banach space are semi-|·|-summands, so no new concept appears by predualization of semi-|·|-idealoids.     

Smooth partitions of unity in preduals of WCG spaces

This paper is based on part of the author's Ph.D. thesis written under the supervision of Professor V. Zizler and has been supported in part by a Province of Alberta Graduate Fellowship     

Optimization and differentiation in banach spaces

We show that some old ideas of Smulian can be used to give another proof of a theorem of Bourgain. We characterize subsets of Banach spaces having the Radon- Nikodym property by means of optimization results.     

Normal structure of banach spaces

GOSSEZ and LAMI DOZO have obtained a sufficient condition for a Banach space with a Schauder basis to have normal structure. This paper generalizes their theorem to obtain a sufficient condition for an arbitrary Banach space to have normal structure. An example is given to show an application of the theorem.Research supported by a Faculty Research Grant of the College of William and Mary.     

Uniform embeddings of metric spaces and of banach spaces into hilbert spaces

It is proved using positive definite functions that a normed spaceX is unifomly homeomorphic to a subset of a Hilbert space, if and only ifX is (linearly) isomorphic to a subspace of aL ₀(μ) space (=the space of the measurable functions on a probability space with convergence in probability). As a result we get thatl _p (respectivelyL _p (0, 1)), 2<p<∞, is not uniformly embedded in a bounded subset of itself. This answers negatively the question whether every infinite dimensional Banach space is uniformly homeomorphic to a bounded subset of itself. Positive definite functions are also used to characterize geometrical properties of Banach spaces. Partially supported by the National Science Foundation, Grant MCS-79-03322. Partially supported by the National Science Foundation, Grant MCS-80-06073.     

Interpolation of several banach spaces

Summary Peetre's K- and J-methods for interpolation are extended to the situation of more than two spaces. The theory developed is applied to interpolation of L_p-spaces with weights and to spaces of Besov and Sobolev type. Entrata in Redazione il 5 luglio 1972.     

Criteria for Eberlein compactness in spaces of continuous functions

Criteria for pointwise relative Eberlein compactness in spaces of continuous maps and in spaces of linear operators are given in terms of countable compactness, Stone-Cech extendability, and interchangeability of double limits.     

On banach spaces whose duals areL 1 spaces

A structure theorem for Banach spaces whose duals areL ₁ spaces, is proved. The research of the second named author has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18 through the European Office of Aerospace Research (OAR) United States Air Force.     

Separable reflexive banach spaces that are universal for the class of the spaces Lp (1 < p < ∞)

Mathematical Notes -     

The preparation theorem on banach spaces

We give a generalization, for smooth Fredholm maps between Banach spaces, of the Preparation Theorem known in finite dimension. As an application we obtain the Prepared Form Theorem which is a basic tool in singularity theory.     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.