ON CONTINUOUS MODULE HOMOMORPHISMS BETWEEN RANDOM LOCALLY CONVEX MODULES

- Journal title : Journal of the Korean Mathematical Society
- Volume 50, Issue 5, 2013, pp.933-944
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2013.50.5.933

Title & Authors

ON CONTINUOUS MODULE HOMOMORPHISMS BETWEEN RANDOM LOCALLY CONVEX MODULES

Zhang, Xia;

Zhang, Xia;

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 -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;()-topology;locally -convex topology;continuous module homomorphisms;

Language

Korean

References

1.

N. Dunford and J. T. Schwartz, Linear Operators, Part I, A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1988.

2.

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.

3.

T. X. Guo, Extension theorems of continuous random linear operators on random domains, J. Math. Anal. Appl. 193 (1995), no. 1, 15-27.

4.

T. X. Guo, Module homomorphisms on random normed modules, Northeast. Math. J. 12 (1996), no. 1, 102-114.

5.

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.

6.

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.

7.

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.

8.

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.

9.

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.

10.

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.

11.

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.

12.

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.

13.

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.

14.

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).

15.

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.

16.

B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier/North-Holland, New York, 1983; Dover Publications, New York, 2005.

17.

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.

18.

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.

19.

X. Zhang, On mean ergodic semigroups of random linear operators, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 4, 53-58.

20.

X. Zhang, On conditional mean ergodic semigroups of random linear operators, J. Inequal. Appl. 150 (2012), 1-10.

21.

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.

22.

S. E. Zhao and T. X. Guo, The random subreflexivity of complete random normed modules, Internat. J. Math. 23 (2012), no. 3, 1-14.