A Minimax Theorem for (L)over-bar0-Valued Functions on Random Normed Modules

We generalize the well-known minimax theorems to (L) over bar (0)-valued functions on random normed modules. We first give some basic properties of an L-0-valued lower semicontinuous function on a randomnormed module under the two kinds of topologies, namely, the (epsilon,lambda)-topology and the locally L-0 -convex topology. Then, we introduce the definition of random saddle points. Conditions for an L-0-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 (L) over bar (0)-valued functions in the framework of random normed modules and random conjugate spaces.