Relations between some basic results derived from two kinds of topologies for a random locally convex module

The purpose of this paper is to exhibit the relations between some basic results derived from the two kinds of topologies (namely the (ε,λ)-topology and the stronger locally L0-convex topology) for a random locally convex module. First, we give an extremely simple proof of the known Hahn–Banach extension theorem for L0-linear functions as well as its continuous variant. Then we give the relations between the hyperplane separation theorems in [D. Filipović, M. Kupper, N. Vogelpoth, Separation and duality in locally L0-convex modules, J. Funct. Anal. 256 (2009) 3996–4029] and a basic strict separation theorem in [T.X. Guo, H.X. Xiao, X.X. Chen, A basic strict separation theorem in random locally convex modules, Nonlinear Anal. 71 (2009) 3794–3804]: in the process we also obtain a very useful fact that a random locally convex module with the countable concatenation property must have the same completeness under the two topologies. As applications of the fact, we prove that most of the previously established principal results of random conjugate spaces of random normed modules under the (ε,λ)-topology are still valid under the locally L0-convex topology, which considerably enriches financial applications of random normed modules.