On L0-convex compactness in random locally convex modules

Abstract

For the study of some typical problems in finance and economics, Žitković introduced convex compactness and gave many remarkable applications. Recently, motivated by random convex optimization and random variational inequalities, Guo, et al. introduced L0-convex compactness, developed the related theory of L0-convex compactness in random normed modules and further applied it to backward stochastic equations. In this paper, we extensively study L0-convexly compact sets in random locally convex modules so that a series of fundamental results are obtained. First, we show that every L0-convexly compact set is complete (hence is also closed and has the countable concatenation property). Then, we prove that any L0-convexly compact set is linearly homeomorphic to a weakly compact subset of some locally convex space, and simultaneously establish the equivalence between L0-convex compactness and convex compactness for a closed L0-convex set. Finally, we establish Tychonoff type, James type and Banach-Alaoglu type theorems for L0-convex compactness, respectively.