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.


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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