[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2013.50.5.933

ON CONTINUOUS MODULE HOMOMORPHISMS BETWEEN RANDOM LOCALLY CONVEX MODULES

Zhang, Xia (Department of Mathematics School of Science Tianjin Polytechnic University)

Publication Information

Journal of the Korean Mathematical Society / v.50, no.5, 2013 , pp. 933-944 More about this Journal

Abstract

Based on the four kinds of theoretical definitions of the continuous module homomorphism between random locally convex modules, we first show that among them there are only two essentially. Further, we prove that such two are identical if the family of $L^0$ -seminorms for the former random locally convex module has the countable concatenation property, meantime we also provide a counterexample which shows that it is necessary to require the countable concatenation property.

Keywords

random locally convex modules; ( ${\varepsilon},{\lambda}$ )-topology; locally $L^0$ -convex topology; continuous module homomorphisms;

Citations & Related Records

Reference

1	M. Z. Wu, The Bishops-Phelps theorem in complete random normed modules endowed with the $({\varepsilon},{\lambda})$ -topology, J. Math. Anal. Appl. 391 (2012), no. 2, 648-952. DOI ScienceOn
2	M. Z. Wu, A further study on the Riemann-intergrability for abstract-valued functions from a closed real interval to a complete random normed module (in Chinese), Sci. Sin. Math. 42 (2012), no. 9, 897-903. DOI
3	X. Zhang, On mean ergodic semigroups of random linear operators, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 4, 53-58. DOI
4	X. Zhang, On conditional mean ergodic semigroups of random linear operators, J. Inequal. Appl. 150 (2012), 1-10.
5	X. Zhang and T. X. Guo, Von Neumann's mean ergodic theorem on complete random inner product modules, Front. Math. China 6 (2011), no. 5, 965-985. DOI ScienceOn
6	S. E. Zhao and T. X. Guo, The random subreflexivity of complete random normed modules, Internat. J. Math. 23 (2012), no. 3, 1-14.
7	S. E. Zhao and G. Shi, A geometric form of the Hahn-Banach extension theorem for $L^0$ linear functions and the Goldstine-Weston theorem in random normed modules (in Chinese), Sci. Sin. Math. 41 (2011), no. 9, 827-836. DOI
8	T. X. Guo, Survey of recent developments of random metric theory and its applications in China. II, Acta Anal. Funct. Appl. 3 (2001), no. 3, 208-230.
9	T. X. Guo, The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani- Smulian theorem in complete random normed modules to stratification structure, Sci. China Ser. A 51 (2008), no. 9, 1651-1663. DOI ScienceOn
10	T. X. Guo, Relations between some basic results derived from two kinds of topologies for a random locally convex module, J. Funct. Anal. 258 (2010), no. 9, 3024-3047. DOI ScienceOn
11	T. X. Guo, Recent progress in random metric theory and its applications to conditional risk measures, Sci. China Math. 54 (2011), no. 4, 633-660. DOI ScienceOn
12	T. X. Guo and S. B. Li, The James theorem in complete random normed modules, J. Math. Anal. Appl. 308 (2005), no. 1, 257-265. DOI ScienceOn
13	T. X. Guo and G. Shi, The algebraic structure of finitely generated $L^0$ (F,K)-modules and the Helly theorem in random normed modules, J. Math. Anal. Appl. 381 (2011), no. 2, 833-842. DOI ScienceOn
14	T. X. Guo and X. Zhang, Stone's representation theorem of a group of random unitary operators on complete complex random inner product modules (in Chinese), Sci. Sin. Math. 42 (2012), no. 3, 181-202. DOI
15	T. X. Guo and S. E. Zhao, On the random conjugate spaces of a random locally convex module, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 4, 687-996. DOI
16	T. X. Guo, S. E. Zhao, and X. L. Zeng, On random convex analysis-the analytic foundation of the module approach to conditional risk measures, arXiv:1210.1848, (2012).
17	T. X. Guo and L. H. Zhu, A characterization of continuous module homomorphisms on random semi-normed modules and its applications, Acta Math. Sin. (Engl. Ser.) 19 (2003), no. 1, 201-208. DOI
18	B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier/North-Holland, New York, 1983; Dover Publications, New York, 2005.
19	N. Dunford and J. T. Schwartz, Linear Operators, Part I, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1988.
20	D. Filipovic, M. Kupper, and N. Vogelpoth, Separation and duality in locally $L^0$ -convex modules, J. Funct. Anal. 256 (2009), no. 12, 3996-4029. DOI ScienceOn
21	T. X. Guo, Extension theorems of continuous random linear operators on random domains, J. Math. Anal. Appl. 193 (1995), no. 1, 15-27. DOI ScienceOn
22	T. X. Guo, Module homomorphisms on random normed modules, Northeast. Math. J. 12 (1996), no. 1, 102-114.
23	T. X. Guo, Some basic theories of random normed linear spaces and random inner product spaces, Acta Anal. Funct. Appl. 1 (1999), no. 2, 160-184.