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The James theorem in complete random normed modules. (English) Zbl 1077.46061

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 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 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 reflexivity given in Theorem 3.1. This is based on the concept of PN-proximality for a subset \(G\) of \(S\), a form of best approximation with respect to the module norm \(\mathcal{X}\). The proof relies on previous results, mainly by the first author.

MSC:

46S50 Functional analysis in probabilistic metric linear spaces
54E70 Probabilistic metric spaces
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