The James theorem in complete random normed modules
type: Item
instance of: scholarly article
title: The James theorem in complete random normed modules
zbMATH Open document ID: 1077.46061
author: Shaobin Li
author: Tiexin Guo
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-06-30T00:00:00Z
review text: Given a \(\sigma\)-finite measure space \((\Omega,\mathcal{A},\mu)\), a random normed (= RN) space is an ordered pair \((S,\mathcal{X)}\), where \(S\) is a linear space over \(\mathbb{K}\) \((=\mathbb{R}\) or \(\mathbb{C})\) and \(\mathcal{X}\) is a mapping from \(S\) into \(L^{+}(\mu)\), the set of equivalence classes of positive measurable real functions. If \(\mathcal{X}(p)=X_p\), it is assumed that, for all \(\alpha\in\mathbb{K}\) and all \(p,q\in S\), (i) \(X_{\alpha p}=| \alpha| \,X_p\), (ii) \(X_{p+q}\leq X_p+X_q\), (iii) \(X_p=0\Rightarrow p=\theta\) (the null vector of \(S\)). If there exists a second mapping \(\ast:L(\mu,\mathbb{K})\times S \to S\) such that (iv) \((S,\ast)\) is a left module over the algebra \(L(\mu,\mathbb{K})\), and (v) for all \(\xi\in L(\mu,\mathbb{K})\) and all \(p\in S\), \(X_{\xi\ast p} =| \xi| \,X_p\), then the triple \((S,\mathcal{X},\ast)\) is called an RN module. An RN module is endowed with a metrizable topology [see \textit{T. Guo}, J. Xiamen Univ., Nat. Sci. 36, No. 4, 499--502 (1997; Zbl 0902.46053)]; \(X_p\) plays the role of the probabilistic norm of \(p\). A sequence \(\{p_n\}\) converges to \(p\in S\) if \(\{X_{p_n-p}\}\) converges to \(0\) in \(\mu\)-measure on every set \(A\in\mathcal{A}\) of finite measure and the module multiplication \(\cdot:L(\mu,\mathbb{K})\times S\to S\) is jointly continuous. In definition 2.2, \(\mu\)-a.e. bounded linear functionals are introduced as well as the dual RN module \((S^*,\mathcal{X}^*,\otimes)\), with suitable definitions of \(\mathcal{X}^*\) and of \(\otimes\); the dual RN module is always complete [see \textit{T. Guo}, Northeast. Math. J. 12, No. 1, 102--114 (1996; Zbl 0858.60012)]. The canonical embedding \(J:S\to S^{**}\), where \(S^{**}\) is the double dual of \(S\), is defined by \(J(p)(f)=f(p)\) and is a measure-preserving module homomorphism; if it is also onto, then \(S\) is said to be reflexive. The main result of this interesting paper is the characterization of random …
reviewed by: Carlo Sempi
Mathematics Subject Classification ID: 46S50
Mathematics Subject Classification ID: 54E70
zbMATH DE Number: 2184083
zbMATH Keywords: random normed modules
zbMATH Keywords: James theorem
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.01.024
OpenAlex ID: W2080735663
cites work: Q4148368
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: Q4489016
cites work: Extension theorems of continuous random linear operators on random domains
cites work: Radon-Nikodým property of conjugate Banach spaces and \(w^*\)-equivalence theorems of \(w^*-\mu\)-measurable functions
cites work: Q4895670
cites work: A note on pointwise best approximation
cites work: Q4368205
cites work: Representation theorems of the dual of Lebesgue-Bochner function spaces
cites work: Proximinality in \(L_ p(S,Y)\)
cites work: Q3336431
cites work: Pointwise best approximation in the space of strongly measurable functions with applications to best approximation in \(L^ p(\mu,X)\)
DOI: 10.1016/J.JMAA.2005.01.024
description: scientific article; zbMATH DE number 2184083