\(p\)-Banach spaces and \(p\)-totally convex spaces. II
type: Item
instance of: scholarly article
title: \(p\)-Banach spaces and \(p\)-totally convex spaces. II
zbMATH Open document ID: 0977.18006
DOI: 10.1023/A:1008600512130
author: Ralf Kemper
published in: Applied Categorical Structures
publication date: 2002-01-17T00:00:00Z
review text: Let \(\mathbb{K}\) be \(\mathbb{R}\) or \(\mathbb{C}\) and let \(p\) be denote a real number with \(0<p\leq 1\) in the following. A \(p\)-absolutely convex set \(C\) (with respect to \(\mathbb{K})\) is a non-empty subset of a \(\mathbb{K}\) vector space such that, for any \(x_i\in C\), \(1\leq i\leq n\), and any \(\alpha_i\in \mathbb{K}\), \(1\leq i\leq n\), with \(\sum^n_{i=1} |\alpha_i |^p\leq 1\), \(\sum^n_{i=1} \alpha_ix_i\in C\) follows. It should be noted that the author here uses a notation different from the one used in the literature: His ``\(p\)-absolutely convex'' is usually called ``absolutely \(p\)-convex''. A \(p\)-absolutely convex space \(C\) is defined as a natural generalization of a \(p\)-absolutely convex set. It is a non-empty set admitting all \(p\)-absolutely convex combinations \[ (\alpha_i\mid 1\leq i\leq n)\in \Omega_{p,fin}: =\left\{(\alpha_\nu \mid 1\leq \nu\leq m)\mid \alpha_\nu \in\mathbb{K}, \sum^m_{\nu=1} |\alpha_\nu |^p\leq 1,\;m\in\mathbb{N} \right\} \] and satisfying the usual equations, but it is not, in general, a subset of a vector space with the induced operations. These structures were introduced and investigated in the first part of the paper by the author [\textit{R. Kemper}, ``\(p\)-Banach spaces and \(p\)-totally convex spaces'', Appl. Categ. Struct. 7, No. 3, 279-295 (1999; Zbl 0949.18002)]. The \(p\)-absolutely convex spaces are equationally defined algebras with \(\Omega_{p,fin}\) as operator domain and, together with the operation preserving mappings, form a category \({\mathcal A}{\mathcal C}_p\). By taking as operator domain \[ \Omega_p: =\left\{(\alpha_i\mid i\in\mathbb{N}) \mid\alpha_i \in\mathbb{K}, \sum^\infty_{i=1} |\alpha_i |^p\leq 1\right\} \] one gets in an analogous way the category \({\mathcal T}{\mathcal C}_p\) of \(p\)-totally convex spaces (with respect to \(\mathbb{K})\) [loc. cit.]. A paradigmatic example of a \(p\)-totally convex space is the closed unit ball \(O_p(E)\) of a \(p\)-Banach space \(E\). …
reviewed by: Dieter Pumplün
Mathematics Subject Classification ID: 18C20
Mathematics Subject Classification ID: 46B99
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 18C05
zbMATH DE Number: 1553290
zbMATH Keywords: totally \(p\)-convex
zbMATH Keywords: \(p\)-absolutely convex
zbMATH Keywords: absolutely \(p\)-convex
zbMATH Keywords: \(p\)-totally convex
zbMATH Keywords: \(p\)-Banach space
zbMATH Keywords: \(p\)-normed vector spaces
MaRDI profile type: Publication
full work available at URL: a:1008600512130
OpenAlex ID: W2010941090
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description: scientific article; zbMATH DE number 1553290