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Journal of Function Spaces and Applications
Volume 2013, Article ID 704251, 10 pages
http://dx.doi.org/10.1155/2013/704251
Research Article

A Minimax Theorem for -Valued Functions on Random Normed Modules

1Elementary Educational College, Capital Normal University, Beijing 100048, China
2Department of Basic Sciences, Hebei Finance University, Baoding 071051, China

Received 9 April 2013; Accepted 25 July 2013

Academic Editor: Pei De Liu

Copyright © 2013 Shien Zhao and Yuan Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We generalize the well-known minimax theorems to -valued functions on random normed modules. We first give some basic properties of an -valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, the ( )-topology and the locally -convex topology. Then, we introduce the definition of random saddle points. Conditions for an -valued function to have a random saddle point are given. The most greatest difference between our results and the classical minimax theorems is that we have to overcome the difficulty resulted from the lack of the condition of compactness. Finally, we, using relations between the two kinds of topologies, establish the minimax theorem of -valued functions in the framework of random normed modules and random conjugate spaces.