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.


46S50 Functional analysis in probabilistic metric linear spaces
54E70 Probabilistic metric spaces
Full Text: DOI


[1] Diestel, J.; Uhl, J. J., Vector Measures, Math. Surveys, vol. 15 (1977), Amer. Math. Soc. · Zbl 0369.46039
[2] Dunford, N.; Schwartz, J. T., Linear Operators, Part I (1957), Interscience: Interscience New York
[3] Guo, T.-X., Survey of recent developments of random metric theory and its applications in China (I), Acta Anal. Funct. Appl., 3, 129-158 (2001) · Zbl 0989.54035
[4] Guo, T.-X., Survey of recent developments of random metric theory and its applications in China (II), Acta Anal. Funct. Appl., 3, 208-230 (2001) · Zbl 0989.54036
[5] Guo, T.-X., Some basic theories of random normed linear spaces and random inner product spaces, Acta Anal. Funct. Appl., 1, 160-184 (1999) · Zbl 0965.46010
[6] Guo, T.-X., Extension theorems of continuous random linear operators on random domains, J. Math. Anal. Appl., 193, 15-27 (1995) · Zbl 0879.47018
[7] Guo, T.-X., The Radon-Nikodým property of conjugate spaces and the \(\omega^\ast-μ\)-equivalence theorem for the \(\omega^\ast-μ\)-measurable functions, Sci. China Ser. A, 39, 1034-1041 (1996) · Zbl 0868.46014
[8] Guo, T.-X., Module homomorphisms on random normed modules, China Northeastern Math. J., 12, 102-114 (1996) · Zbl 0858.60012
[9] Guo, T.-X.; You, Z.-Y., A note on pointwise best approximation, J. Approx. Theory, 93, 344-347 (1998) · Zbl 0912.41016
[10] Guo, T.-X., A characterization for a complete random normed module to be random reflexive, J. Xiamen Univ. Natur. Sci., 36, 499-502 (1997), (in Chinese) · Zbl 0902.46053
[11] Guo, T.-X., Representation theorems of the dual of Lebesgue-Bochner function spaces, Sci. China Ser. A, 43, 234-243 (1999) · Zbl 0959.46024
[12] Light, W. A., Proximinality in \(L_p(S, Y)\), Rocky Mountain J. Math., 1, 251-259 (1989) · Zbl 0722.46013
[13] Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (1983), Elsevier/North-Holland: Elsevier/North-Holland New York · Zbl 0546.60010
[14] Schweizer, B., Commentary on probabilistic geometry, (Schweizer, B.; Sklar, A.; Sigmund, K.; etal., Karl Menger, Selecta Math., vol. 2 (2003), Springer-Verlag: Springer-Verlag Wien), 409-432
[15] You, Z.-Y.; Guo, T.-X., Pointwise best approximation in the space of strongly measurable functions with applications to best approximation in \(L^p(\mu, X)\), J. Approx. Theory, 78, 314-320 (1994) · Zbl 0808.41024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.