A characterization of continuous module homomorphisms on random semi-normed modules and its applications
type: Item
instance of: scholarly article
title: A characterization of continuous module homomorphisms on random semi-normed modules and its applications
zbMATH Open document ID: 1027.60069
author: Linhu Zhu
author: Tiexin Guo
publication date: 2003-09-25T00:00:00Z
review text: The authors prove the following main theorem: Suppose \((S,\{\chi^d\}_{d\in D})\) is a random semi-normed module with base \((\Omega,{\mathcal A},\mu)\) over \(K\), and \(f: S\to L(\mu, K)\) is a linear mapping. Then the following statements are equivalent: (1) \(f\) is a continuous module homomorphism; (2) there exist a countable partition \(\{A_i\mid i\in\mathbb{N}\}\) of \(\Omega\) to \({\mathcal A}\), a sequence \(\{C_i\mid i\in\mathbb{N}\}\) in \(L^+(\mu)\) with the \(\mu\)-essential support of \(C_i\) contained in \(A_i\) for each \(i\in\mathbb{N}\), and a countable subfamily \(\{H_i\mid i\in\mathbb{N}\}\) of \({\mathcal F}(D)\) such that \(|f(p)|\leq \sum^\infty_{i=1} C_i\cdot X^{H_i}_p\), \(\forall p\in S\), where \({\mathcal F}(D)\) stands for the collection of all finite subsets of \(D\), and \(\chi^{H_i}: S\to L^+(\mu)\) is defined by \(X^{H_i}_p= \chi^{H_i}(p)= \sum_{d\in H_i} X^d_p\), \(\forall i\geq 1\) and \(p\in S\), or by \(X^{H_i}_p= \sum_{d\in H_i} X^d_p\), \(\forall i\geq 1\) and \(p\in S\); (3) there exists a continuous random seminorm \(F\) on \(S\) satisfying \(F(\xi\cdot p)= \xi\cdot F(p)\), \(\forall\xi\in L^+(\mu)\) and \(p\in S\), such that \(|f(p)|\leq F(p)\), \(\forall p\in S\).
reviewed by: Stanisław Wedrychowicz
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 60G99
Mathematics Subject Classification ID: 46A22
Mathematics Subject Classification ID: 46A16
zbMATH DE Number: 1921190
MaRDI profile type: Publication
full work available at URL: s10114-002-0210-x
OpenAlex ID: W2011758788
cites work: Shrinkable neighborhoods in Hausdorff linear spaces
cites work: Q3227025
cites work: Radon-Nikodým property of conjugate Banach spaces and \(w^*\)-equivalence theorems of \(w^*-\mu\)-measurable functions
cites work: Representation theorems of the dual of Lebesgue-Bochner function spaces
cites work: Q4489016
cites work: Survey of recent developments of random metric theory and its applications in China. I
cites work: Survey of recent developments of random metric theory and its applications in China. II
cites work: Q3254057
DOI: 10.1007/S10114-002-0210-X
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 1921190