Joe Diestel

• University of Granada

We continue the study of the w-right and strong* topologies in general Banach spaces started in 36, 37 and 35. We show that in L1(μ)-spaces the w-right convergence of sequences admits a simpler control. Some considerations about these topologies will be contemplated in the particular cases of C*-algebras and JB*-triples in connection with summing o...

In this part of our exposition and expansion of Grothendieck's work on the metric theory of tensor products we look at the behaviour of the injective and surjective tensor norms in the case where one of the component spaces is a classical sequence space as exposed in Grothendieck's “Sur certaines classes de suites dans les espaces de Banach, et le...

In this continuation of our exposition and expansion of Grothendieck's 'Résumé,' we broach the subject of bilinear forms and operators of type α, where α is a tensor norm, and follow that up with the closely related subject of α-nuclearity.

In this continuation of our multiseries on Grothendieck's 'Resume', we look at the projective and injective properties of general tensor norms. We also introduce the concepts of left and right injectivity as well as left and right projectivity.

In this paper we discuss the problem of when the projective tensor product of two Banach spaces has the Radon-Nikodym property. We give a detailed exposition of the famous examples of Jean Bourgain and Gilles Pisier showing that there are Banach spaces X and Y such that each has the Radon-Nikodym property but for which their projective tensor produ...

This note improves two previous results of the second author. They turn out to be special cases of our main theorem which states: A Banach space X has the property that the strong closure of every abstractly -complete Boolean algebra of projections in X is Bade complete if and only if X does not contain a copy of the sequence space' 1 .

Let $X$ and $Y$ be Banach spaces such that $X$ has a boundedly complete basis. Then $X\hat{\otimes}Y$, the projective tensor product of $X$ and $Y$, has the Radon-Nikodym property (resp.~the analytic Radon-Nikodym property, the near Radon-Nikodym property, contains no copy of $c_0$) if and only if $Y$ has the same property.

This chapter discusses about the Riesz theorem. The use of the Riesz theorem that is used in tandem with the Hahn–Banach theorem is examined in the chapter. One of the first applications of the Riesz theorem is in characterizing weakly convergent sequences in metric spaces. The partnership of the Riesz Theorem with various forms of the Hahn–Banach...

In this continuation of our multiseries on Grothendieck's 'Résumé', we look at the special place of C(K)-spaces and L1-spaces in the general metric theory of tensor products.

A sequential description of the projective tensor product lp⊗X, is given. This description allows us to show the inclusion lp⊗X, into lp⊗X, is a semi-embedding. As a consequence, if X has the Radon-Nikodym property so does lp⊗X, A discussion of difficulties with L(0, 1) ⊗X, follows.

Throughout the literature on optimal control in Banach spaces, hypotheses like “separable and reflexive” are frequently encountered. In this note we consider one such case, studied by Peichl and Schappacher. Using techniques from Banach space theory and the theory of vector measures, we show how to remove the hypothesis of reflexivity and translate...

Many fundamental processes in analysis are best understood by studying and comparing the summability of series in various modes of convergence. This text provides the beginning graduate student, one with basic knowledge of real and functional analysis, with an account of p-summing and related operators. The account is panoramic, with detailed expos...

Let (Ω,Σ,μ) be a finite measure space and X a Banach space. Denote by L 1 (μ, X ) the Banach space of (equivalence classes of) μ-strongly measurable X -valued Bochner integrable functions f :Ω→ X normed by The problem of characterizing the relatively weakly compact subsets of L 1 (Ω, X ) remains open. It is known that for a bounded subset of L 1 (...

Undecima Escuela Venezolana de Matemáticas Incluye bibliografía

XVII Escuela Venezolana de Matemáticas. IV Escuela Matemática de América Latina y El Caribe, Mérida, Venezuela, 5 al 11 Septiembre de 2004 Incluye bibliografía