Geometry and Gâteaux smoothness in separable Banach spaces
type: Item
instance of: scholarly article
title: Geometry and Gâteaux smoothness in separable Banach spaces
zbMATH Open document ID: 1277.46006
author: Petr Hájek
author: V. Montesinos
author: V. Zizler
published in: Operators and Matrices
publication date: 2012-09-14T00:00:00Z
Mathematics Subject Classification ID: 46B03
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 46B22
Mathematics Subject Classification ID: 46G05
zbMATH DE Number: 6082482
zbMATH Keywords: Gâteaux differentiable norms
zbMATH Keywords: extreme points
zbMATH Keywords: Radon-Nikodým property
zbMATH Keywords: superreflexive spaces
zbMATH Keywords: Hilbertian spaces
zbMATH Keywords: differentiability of norms
zbMATH Keywords: bump function
zbMATH Keywords: modulus of smoothness
zbMATH Keywords: preserved extreme points
MaRDI profile type: Publication
full work available at URL: oam-06-15
OpenAlex ID: W2046985877
DOI: 10.7153/OAM-06-15
review text: The paper is a survey on the interplay between differentiability of norms of real Banach spaces with the geometry of Banach spaces. Proofs of the stated theorems are sometimes omitted or sketched and many open problems are mentioned. We refer for the unexplained notions to the paper. The paper is divided into several sections.NEWLINENEWLINELet \((X,\|\cdot\|)\) be a Banach space. By a norm on \(X\), it is always meant an equivalent norm on \(X\).NEWLINENEWLINEThe first section studies pointwise directional Hölder derivatives. It is known that every separable Banach space can be (equivalently) renormed by a uniformly Gâteaux differentiable norm. The situation is different if it is required that the norm has directional Hölder derivative (pointwise or uniform). The following result summarizes some sufficient conditions for a space to be superreflexive (a space is superreflexive if it admits a uniformly Fréchet differentiable norm).NEWLINENEWLINELet \(X\) be a Banach space. Then each one of the following conditions implies that \(X\) is superreflexive. {\parindent=0.7cm\begin{itemize}\item[(i)] The space has the Radon-Nikodým property and admits a continuous bump function with pointwise directional Hölder derivative. \item[(ii)] Both \(X\) and \(X^*\) admit continuous bump functions with pointwise directional Hölder derivative. \item[(iii)] The space \(X\) admits a locally uniformly rotund norm with pointwise directional Hölder derivative on the sphere. \item[(iv)] The space \(X\) admits a bounded bump function with uniformly directional Hölder derivative. NEWLINENEWLINE\end{itemize}} Due to Pisier's result that every superreflexive Banach space has a norm with modulus of smoothness of power type [\textit{G. Pisier}, Isr. J. Math. 20, 326--350 (1975; Zbl 0344.46030)] and several other results, each of the conditions above characterizes superreflexive spaces.NEWLINENEWLINEThe second section of the paper focuses on the second-order Gâteaux differentiability. It is stated…
reviewed by: J. Spurný
description: scientific article; zbMATH DE number 6082482