## 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

### Keywords:

locally convex module; Hahn–Banach theorem; duality

### Citations:

Zbl 0477.46017; Zbl 0514.03032; Zbl 1087.03032; Zbl 1137.49002
Full Text:

### References:

 [1] Brannath, W.; Schachermayer, W., A bipolar theorem for subsets of $$L_+^0(\Omega, \mathcal{F}, P)$$, (), 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 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 [5] T.X. Guo, The theory of probabilistic metric spaces with applications to random functional analysis, Master’s thesis, Xi’an Jiaotong University (China), 1989 [6] T.X. Guo, Random metric theory and its applications, PhD thesis, Xi’an Jiaotong University (China), 1992 [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 $$\operatorname{w}^*$$-equivalence theorem for $$\operatorname{w}^*$$-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 [17] T.X. Guo, The theory of module homomorphisms in complete random inner product modules and its applications to Skorohod’s random operator theory, submitted for publication [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 [27] M. Kupper, N. Vogelpoth, Complete $$L^0$$-normed modules and automatic continuity of monotone convex functions, Working paper Series No. 10, Vienna Institute of Finance, 2008 [28] Schweizer, B.; Sklar, A., Probabilistic metric spaces, (2005), Dover Publications Mineola, New York, reissued by · Zbl 0546.60010 [29] Skorohod, A.V., Random linear operators, (1984), 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
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