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
Full Text: DOI arXiv


[1] Brannath, W.; Schachermayer, W., A bipolar theorem for subsets of \(L_+^0(\Omega, F, P)\), (Séminaire de Probabilités XXXIII. Séminaire de Probabilités XXXIII, Lecture Notes in Math., vol. 1709 (1999), Springer), 349-354 · Zbl 0957.46020
[2] Breckner, W. W.; Scheiber, E., A Hahn-Banach extension theorem for linear mappings into ordered modules, Mathematica, 19, 42, 13-27 (1977) · Zbl 0396.46002
[3] Dunford, N.; Schwartz, J. T., Linear Operators (I) (1957), Interscience: Interscience New York
[4] Filipović, D.; Kupper, M.; Vogelpoth, N., Separation and duality in locally \(L^0\)-convex modules, J. Funct. Anal., 256, 3996-4029 (2009) · Zbl 1180.46055
[7] Guo, T. X., Extension theorems of continuous random linear operators on random domains, J. Math. Anal. Appl., 193, 1, 15-27 (1995) · Zbl 0879.47018
[8] Guo, T. X., The Radon-Nikodým property of conjugate spaces and the \(w^*\)-equivalence theorem for \(w^*\)-measurable functions, Sci. China Ser. A, 39, 1034-1041 (1996) · Zbl 0868.46014
[9] Guo, T. X., Module homomorphisms on random normed modules, Chinese Northeast. Math. J., 12, 102-114 (1996) · Zbl 0858.60012
[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) · Zbl 0902.46053
[11] Guo, T. X., Some basic theories of random normed linear spaces and random inner product spaces, Acta Anal. Funct. Appl., 1, 2, 160-184 (1999) · Zbl 0965.46010
[12] Guo, T. X., Representation theorems of the dual of Lebesgue-Bochner function spaces, Sci. China Ser. A, 43, 234-243 (2000) · Zbl 0959.46024
[13] 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
[14] 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
[15] Guo, T. X., Several applications of the theory of random conjugate spaces to measurability problems, Sci. China Ser. A, 50, 737-747 (2007) · Zbl 1128.46031
[16] Guo, T. X., The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Šmulian theorem in complete random normed modules to stratification structure, Sci. China Ser. A, 51, 1651-1663 (2008) · Zbl 1167.46049
[18] Guo, T. X.; Chen, X. X., Random duality, Sci. China Ser. A, 52, 2084-2098 (2009) · Zbl 1193.46048
[19] Guo, T. X.; Li, S. B., The James theorem in complete random normed modules, J. Math. Anal. Appl., 308, 257-265 (2005) · Zbl 1077.46061
[20] Guo, T. X.; Peng, S. L., A characterization for an \(L(\mu, K)\)-topological module to admit enough canonical module homomorphisms, J. Math. Anal. Appl., 263, 580-599 (2001) · Zbl 1014.46018
[21] Guo, T. X.; Xiao, H. X., A separation theorem in random normed modules, J. Xiamen Univ. Natur. Sci., 42, 270-274 (2003) · Zbl 1046.46020
[22] Guo, T. X.; Xiao, H. X.; Chen, X. X., A basic strict separation theorem in random locally convex modules, Nonlinear Anal., 71, 3794-3804 (2009) · Zbl 1184.46068
[23] Guo, T. X.; You, Z. Y., The Riesz’s representation theorem in complete random inner product modules and its applications, Chinese Ann. Math. Ser. A, 17, 361-364 (1996)
[24] Guo, T. X.; Zhu, L. H., A characterization of continuous module homomorphisms on random seminormed modules and its applications, Acta Math. Sin. (Engl. Ser.), 19, 1, 201-208 (2003) · Zbl 1027.60069
[25] Ionescu Tulcea, A.; Ionescu Tulcea, C., On the lifting property (I), J. Math. Anal. Appl., 3, 537-546 (1961) · Zbl 0122.11604
[26] Ionescu Tulcea, A.; Ionescu Tulcea, C., On the lifting property (II), J. Math. Mech., 11, 5, 773-795 (1962) · Zbl 0122.11701
[28] Schweizer, B.; Sklar, A., Probabilistic Metric Spaces (2005), Elsevier/North-Holland: Elsevier/North-Holland New York: Dover Publications: Elsevier/North-Holland: Elsevier/North-Holland New York: Dover Publications Mineola, New York, reissued by · Zbl 0546.60010
[29] Skorohod, A. V., Random Linear Operators (1984), D. Reidel Publishing Company: D. Reidel Publishing Company Holland
[30] Vuza, D., The Hahn-Banach theorem for modules over ordered rings, Rev. Roumaine Math. Pures Appl., 9, 27, 989-995 (1982) · Zbl 0505.06010
[31] Wagner, D. H., Survey of measurable selection theorems, SIAM J. Control Optim., 15, 859-903 (1977) · Zbl 0407.28006
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.