A Minimax Theorem forL-0-Valued Functions on Random Normed Modules

We generalize the well-known minimax theorems toL¯0-valued functions on random normed modules. We first give some basic properties of anL0-valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, the (ε,λ)-topology and the locallyL0-convex topology. Then, we introduce the definition of random saddle points. Conditions for anL0-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 ofL¯0-valued functions in the framework of random normed modules and random conjugate spaces.

THE RANDOM SUBREFLEXIVITY OF COMPLETE RANDOM NORMED MODULES

Combining respective advantages of the (ε, λ)-topology and the locally L0-convex topology we first prove that every complete random normed module is random subreflexive under the (ε, λ)-topology. Further, we prove that every complete random normed module with the countable concatenation property is also random subreflexive under the locally L0-convex topology, at the same time we also provide a counterexample which shows that it is necessary to require the random normed module to have the countable concatenation property.