Banach Space Theory

Abstract. We introduce the notions of strong asymptotic uniform smoothness and convexity. We show that the injective tensor product of strongly asymptotically uniformly smooth spaces is asymptotically uniformly smooth. This applies in particular to uniformly smooth spaces admitting a monotone FDD, extending a result by Dilworth, Kutzarova, Randrianarivony, Revalski and Zhivkov [9]. Our techniques also provide a characterisation of Orlicz functions M, N such that the space of compact operators K(h M , h N ) is asymptotically uniformly smooth. Finally we show that K(X, Y ) is not strictly convex whenever X and Y are at least two-dimensional, which extends a result by Dilworth and Kutzarova [7].

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“…Now one can follow the same steps as in the proof of Gurarii's theorem (see, e.g. [12,Lemma 9.26]) to get the statement.…”

Section: On Strongly Auc and Strongly Aus Spacesmentioning

confidence: 98%

“…On the other hand, assume that E is monotone and X is strongly AUC with respect to E. We will argue as in [12,Lemma 9.27]. By Proposition 3.5, F = span{(P E n ) * X * : n ∈ N} is strongly AUS with respect to E * .…”

Section: On Strongly Auc and Strongly Aus Spacesmentioning

confidence: 99%

On strong asymptotic uniform smoothness and convexity

García-Lirola

Raja

2017

Rev Mat Complut

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“…Finally, in the fifth section we investigate sets which are dentable with respect to a metric defined on it, including the proof of Theorem 1.5. Throughout the paper C will denote a closed convex subset of a Banach space X and M will denote a metric space with a metric d. Our notation is standard and will normally follow the books [9] and [3].…”

Section: Theorem 12 Let C ⊂ X Be a Closed Convex Set If M Is A Vecmentioning

confidence: 99%

“…It is related to the differentiation of Lipschitz maps (Aronszajn-Christensen-Mankiewicz's theorem), the extremal structure of convex sets (exposed points), representation theory without compactness (Edgar's theorem), representation of dual function spaces, optimization theory (Stegall's variational principle), etc. The interested reader in RNP, theory and applications, is addressed to [2,5,8,9].…”

Section: Introductionmentioning

confidence: 99%

Maps with the Radon–Nikodým Property

García-Lirola

Raja

2017

Set-Valued Var. Anal

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Abstract. We study dentable maps from a closed convex subset of a Banach space into a metric space as an attempt of generalize the Radon-Nikodým property to a "less linear" frame. We note that a certain part of the theory can be developed in rather great generality. Indeed, we establish that the elements of the dual which are "strongly slicing" for a given uniformly continuous dentable function form a dense G δ subset of the dual. As a consequence, the space of uniformly continuous dentable maps from a closed convex bounded set to a Banach space is a Banach space. However some interesting applications, as Stegall's variational principle, are no longer true beyond the usual hypotheses, sending us back to the classical case. Moreover, we study the connection between dentability and approximation by delta-convex functions for uniformly continuous functions. Finally, we show that the dentability of a set is closely related with the dentability of delta-convex maps defined on it.

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“…a complex Banach space which can be described as Y + iY and with a norm z = Re z + i Im z = sup t∈ [0,2π] cos(t) Re z + sin(t) Im z ; see e.g. [FHHMZ,Section 2.1]. Clearly, R n is isomorphic to C n .…”

mentioning

confidence: 99%

A remark on the approximation theorems of Whitney and Carleman-Scheinberg

Johanis¹

2015

Commentationes Mathematicae Universitatis Carolinae

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Banach Space Theory

Cited by 386 publications

References 0 publications

On strong asymptotic uniform smoothness and convexity

On strong asymptotic uniform smoothness and convexity

Maps with the Radon–Nikodým Property

A remark on the approximation theorems of Whitney and Carleman-Scheinberg

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