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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 operators. We also study (sequential) w-right-norm and strong*-norm continuity for holomorphic mappings. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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. Mathematics Subject Classification (2000): 47B10, 46B28, 46E15, 46E30 Key words: Injective and projective tensor norm, left injective, right injective, left projective, right projective. Quaestiones Mathematicae 26(2003), 479–499

In this continuation of our multiseries on Grothendieck's 'Resume', we look at the special place of C(K)-spaces and L 1-spaces in the general metric theory of tensor products.

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 theorem de Dvoretzky-Rogers ”. Mathematics Subject Classification (2000) : 46B28, 46B45, 47B07. Key words: Weakly p -summable sequences; absolutely p -summable sequences; projective tensor product; injective tensor product; compact operators. Quaestiones Mathematicae 25 (2002), 95-118

In this continuation of our exposition and expansion of Grothendieck's ‘Resume,' 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. Mathematics Subject Classification (2000) : 46B28, 46B07, 46B10. Key words: α-forms;α-integral operators; (metric) accessibility; α-nuclear forms (operators). Quaestiones Mathematicae 25 (2002), 73-94

This paper presents the first of a multi-part series of papers on the metric theory of tensor products according to Grothendieck's “Resume de la theorie metrique des produits tensoriels topologiques” It contains the basics on tensor norms: a discussion of the special character of the injective and the projective norms, fundamental operations on tensor norms, extension of tensor norms to spaces of infinite dimensions. Mathematics Subject Classification (2000): 46B28, 46B07, 46B04. Key words: Reasonable crossnorms; projective norm; injective norm; integral bilinear form; tensor norm; (metric) accessibility. Quaestiones Mathematicae 25 (2002), 37-72

unavailable at this time... Mathematics Subject Classification (2000): 46A, 46B, 46E, 47B. Keywords: tensor, Banach, banach space, tensor product, projective norm, greatest crossnorm, semi-embedding, Radon-Nikodym property, absolutely p-summable sequence, strongly p-summable sequence, topological linear spaces, normed linear spaces, Banach lattices, linear function spaces, duals, function algebras, linear operators, product, Banach spaces Quaestiones Mathematicaes 24 (4) 2001, 519–533

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 the problem of controls to one about the strong continuity of an adjoint semigroup on the positive real axis.

(1992). SOME OBSERVATIONS ABOUT THE SPACE OF WEAKLY CONTINUOUS FUNCTIONS FROM A COMPACT SPACE INTO A BANACH SPACE. Quaestiones Mathematicae: Vol. 15, No. 4, pp. 415-425.

LetX and Y be Banach spaces such thatX has a boundedly complete basis. Then X⊗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 c0) if and only if Y has the same property. 1. Preliminaries Throughout this paper G will denote a compact metrizable abelian group, B(G) is the σ-algebra of Borel subsets of G, and λ is normalized Haar measure on G. The dual group of G will be denoted by Γ. Let X be a real or complex Banach space. We denote by L1(G,X) (respectively, L∞(G,X)) the Banach space of (all equivalence classes of) λ-Bochner integrable functions on G with values in X (respectively, (all equivalence classes of) λ-measurable X-valued functions that are essentially bounded). If μ is a countably additive X-valued measure on B(G), we say that it is of bounded variation if sup ∑ A∈π ‖μ(A)‖

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 theorem is the most powerful in all abstract analysis. The most basic form of the Hahn–Banach theorem extending bounded linear functional from a subspace of a Banach space to the whole space without changing the norm is analyzed in the chapter. The Banach's version of the extension theorem is applied wherein a linear functional that is dominated by a positively homogeneous, subadditive functional is extended to the whole space with domination remaining. One remarkable consequence of Choquet's theorem is the characterization of weakly null sequences in Banach spaces. It is shown that between Hilbert spaces, the 2-summing operators and Hilbert–Schmidt operators are precisely the same, while the trace-class coincides with the nuclear operators.

This chapter explains the theory of operator ideals that was born in 1941 with the observation of Calkin. Schatten and von Neumann actually dealt with more general classes of operators; the so-called cross-spaces of linear transformations. They phrased their results in the language of direct (now tensor) products of the Banach spaces. To stress the analogy with operator ideals, Gothic bold lowercase letters will be used to denote sequence ideals. The local theory of Banach spaces deals with those properties that can be expressed in terms of finite dimensional spaces. The most usual way is to consider the quantitative relations that are assumed to hold uniformly for any choice of n elements and/or functionals. Grothendieck's theorem has many applications in Banach space geometry. Determining precisely when a concrete operator of prescribed form belongs to a given ideal often provides considerable enlightenment about the operator, its domain and codomain, and the ideal.

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 expositions of the core results and highly non-trivial applications to, for example, harmonic analysis, probability and measure theory, and operator theory. Graduate students and researchers from real, complex and functional analysis, and probability theory will benefit from this account.