Geometry of product spaces
type: Item
instance of: scholarly article
title: Geometry of product spaces
zbMATH Open document ID: 1486.46018
author: M. Veena Sangeetha
published in: Journal of Mathematical Analysis and Applications
publication date: 2021-07-09T00:00:00Z
review text: Given a nonempty set \(S\), we denote by \(\mathbb R ^ S\) the set of all real valued functions on \(S\). Let \( \| \cdot \|_E\) be a norm on a linear subspace \(E\) of \(\mathbb R ^S\). We say that the normed space \(E\) is a full function space over \(S\) if, for any \(f \in E\) and \(g: S \rightarrow\mathbb R\) such that \(|g(s)| \leq |f(s)|\) for all \(s \in S\), it is true that \(g \in E\) and \(\| g \|_E \leq \| f \|_E\). Suppose that \(E\) is a full function space over a nonempty set \(S\) and \(\{ X_s \}_{s \in S}\) is a collection of normed spaces. For \(x \in \prod_{s \in S} X_s\), let \(E_x(s) = \| x(s)\|\) for all \(s \in S\). If \(E_x \in E\), let \(\|x\| = \|E_x\|_E\). This function defined on \(\{ x \in \prod_{s\in S} X_s : E_x \in E\}\) is a norm. The resulting normed space is said to be the \(E\)-direct sum of \(\{X_s\}_{s\in S}\) and is denoted by \(\mathcal{P}_E\{X_s\}_{s\in S}\). The standard \(p\)-direct sum (\(1\leq p\leq \infty\)) of normed spaces is an example of an \(E\)-direct sum with \(E = \ell_p\). The author studies the inheritance of some rotundity properties. Let \(X\) be a normed space with unit sphere \(S_X\). We denote the linear and convex spans of a subset \( A\) of \(X\) by \(\text{sp}\, A\) and \(\text{co}\, A\), respectively. Let \(k \in \mathbb N\). Then \(X\) is said to be \(k\)-rotund if, for any affinely independent \(x_1, \dotsc, x_{k+1} \in S_X\) \[ \frac{1}{k+1} \left\|\sum_{i=1}^{k+1} x_i \right\| < 1.\] \(X\) is said to be \(k\)-uniformly rotund if, for every \(\varepsilon > 0\), \[ \inf \left\{ 1-\frac{1}{k+1} \left\|\sum_{i=1}^{k+1} x_i \right\|: x_1, \dotsc, x_{k+1} \in S_X,\ V(x_1, \dotsc, x_{k+1}) \geq \varepsilon \right\} > 0,\] where \(\frac{1}{k!} V (x_1, \dotsc, x_{k+1})\) is the \(k\)-dimensional volume of \(\operatorname{co}\{x_1, \dotsc, x_{k+1}\}\). Then \(X\) is said to be \(k\)-uniformly rotund relative to every \(k\)-dimensional subspace if, for every \(k\)-dimensional subspace \(Y\) of \(X\) and every…
reviewed by: Simon Lücking
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 7370494
zbMATH Keywords: product spaces
zbMATH Keywords: full function spaces
zbMATH Keywords: $k$-uniform rotundity
zbMATH Keywords: $k$-uniform rotundity relative to every $k$-dimensional subspace
zbMATH Keywords: $k$-rotundity
MaRDI profile type: Publication
full work available at URL: j.jmaa.2021.125285
OpenAlex ID: W3157187555
cites work: Absolute and monotonic norms
cites work: Multi-dimensional volumes, super-reflexivity and normal structure in Banach spaces
cites work: Q5753024
cites work: Uniformly Convex Spaces
cites work: Some more uniformly convex spaces
cites work: Uniform convexity. III
cites work: Q3213662
cites work: On convexity properties of \(\psi\)-direct sums of Banach spaces.
cites work: Non-squareness and uniform non-squareness of \(Z\)-direct sums
cites work: Complex strict convexity of absolute norms on \(\mathbb {C}^n\) and direct sums of Banach spaces
cites work: Q3393934
cites work: Q3843864
cites work: Multi-dimensional volumes and moduli of convexity in Banach spaces
cites work: Uniform non-squareness of ψ-direct sums of Banach spaces X ⊕_ψ Y
cites work: Q3558969
cites work: Q4106341
cites work: Local approach to monotonicity properties in \(\psi\)-direct sums of Köthe spaces
cites work: On semi-uniform Kadec-Klee Banach spaces
cites work: Q5805122
cites work: Q5595759
cites work: Banach Spaces that are Uniformly Rotund in Weakly Compact Sets of Directions
cites work: Products of Banach spaces that are uniformly rotund in every direction
cites work: Q4713040
cites work: A Generalization of Uniformly Rotund Banach Spaces
cites work: Strict convexity of absolute norms on \(\mathbb{C}^2\) and direct sums of Banach spaces
cites work: GEOMETRIC AND FIXED POINT PROPERTIES IN PRODUCTS OF NORMED SPACES
cites work: Q5160305
cites work: Q5026428
cites work: Normal structure and invariance of Chebyshev center under isometries
cites work: Q4557936
cites work: Q3496768
DOI: 10.1016/J.JMAA.2021.125285
description: scientific article; zbMATH DE number 7370494