A comprehensive connection between the basic results and properties derived from two kinds of topologies for a random locally convex module

TL;DR

This paper bridges the gap between two topologies in random locally convex modules, simplifying proofs, establishing key relations, and extending deep results to enhance financial applications.

Contribution

It provides a simple proof of the Hahn-Banach theorem, relates separation theorems under two topologies, and shows deep results remain valid under the stronger topology, enriching financial applications.

Findings

Equivalence of completeness under two topologies for modules with the countable concatenation property

Deep results of random conjugate spaces are valid under the locally L^0-topology

Simplified proof of the Hahn-Banach extension theorem for L^0-linear functions

Abstract

The purpose of this paper is to make a comprehensive connection between the basic results and properties derived from the two kinds of topologies (namely the $(\epsilon,\lambda)-$topology introduced by the author and the stronger locally $L^{0}-$convex topology recently introduced by Filipovi$\acute{c}$ et. al) for a random locally convex module. First, we give an extremely simple proof of the known Hahn-Banach extension theorem of $L^{0}-$linear functions as well as its continuous variants. Then we give the essential relations between the hyperplane separation theorems in [Filipovi$\acute{c}$ et. al, J. Funct. Anal.256(2009)3996--4029] and a basic strict separation theorem in [Guo et. al, Nonlinear Anal. 71(2009)3794--3804]: in the process obtain a useful and surprising fact that a random locally convex module with the countable concatenation property must have the same completeness under the two topologies! Based on the relation between the two kinds of completeness, we go on to present the central part of this paper: we prove that most of the previously established deep results of random conjugate spaces of random normed modules under the $(\epsilon,\lambda)-$topology are still valid under the locally $L^{0}-$convex topology, which considerably enriches financial applications of random normed modules.