A Minimax Theorem for L-0-Valued Functions on Random Normed Modules

Abstract EN

We generalize the well-known minimax theorems to L¯0-valued functions on random normed modules.

We first give some basic properties of an L0-valued lower semicontinuous function on a random normed module under the two kinds of topologies, namely, the (ε,λ)-topology and the locally L0-convex topology.

Then, we introduce the definition of random saddle points.

Conditions for an L0-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¯0-valued functions in the framework of random normed modules and random conjugate spaces.