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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 [S. S. 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, E. I. Gordon [Sov. Math., Dokl. 23, 579–582 (1981); translation from Dokl. Akad. Nauk SSSR 258, 777–780 (1981; Zbl 0514.03032)], A. G. Kusraev and S. S. 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.

MSC:

46S50 Functional analysis in probabilistic metric linear spaces
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
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