## 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

### Keywords:

random normed modules; James theorem

### Citations:

Zbl 0902.46053; Zbl 0858.60012
Full Text:

### References:

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