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Relations between some basic results derived from two kinds of topologies for a random locally convex module. (English) Zbl 1198.46058
The author discusses the Hahn--Banach theorem and relevant duality issues in the framework of a module over the ring of real measurable functions. More general results of the sort are readily available in [{\it S. S.\thinspace Kutateladze}, Sib. Math. J. 22, 575--583 (1982); translation from Sib. Mat. Zh. 22, 118--128 (1981; Zbl 0477.46017)]. The model-theoretic nature of convex analysis in modules was disclosed later within Boolean valued analysis; see, for instance, {\it E. I.\thinspace Gordon} [Sov. Math., Dokl. 23, 579--582 (1981); translation from Dokl. Akad. Nauk SSSR 258, 777--780 (1981; Zbl 0514.03032)], {\it A. G.\thinspace Kusraev} and {\it S. S.\thinspace Kutateladze} [“Introduction to Boolean-valued analysis” (Russian) (Moskva: Nauka) (2005; Zbl 1087.03032), “Subdifferential calculus. Theory and applications” (Russian) (Moskva: Nauka) (2007; Zbl 1137.49002)]. The author also addresses two types of completeness in the locally convex modules under consideration.

46S50Functional analysis in probabilistic metric linear spaces
46H25Normed modules and Banach modules, topological modules
46A22Theorems of Hahn-Banach type; extension and lifting of functionals and operators
Full Text: DOI
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