On characterization of locally $L^0$-convex topologies induced by a family of $L^0$-seminorms

Abstract

The purpose of this paper is to provide a characterization of the topological $L^0$-modules whose topology is induced by a family of $L^0$-seminorms using the gauge function for $L^0$-modules. Taking advantage of these ideas we will give a counterexample of a locally $L^0$-convex module whose topology is not induced by a family of $L^0$-seminorms.

References in corpus (1)

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

Cited by in corpus (4)

• A counterexample shows that not every locally $L^0$--convex topology is necessarily induced by a family of $L^0$--seminorms
• Versions of Eberlein-Šmulian and Amir-Lindenstrauss theorems in the framework of conditional sets
• A random version of Mazur's lemma
• Random convex analysis (I): separation and Fenchel-Moreau duality in random locally convex modules