On the Random Conjugate Spaces of a Random Locally Convex Module

Details On the Random Conjugate Spaces of a Random Locally Convex Module On the Random Conjugate Spaces of a Random Locally Convex Module

2011-03-16

arxiv.org/abs/1103.3127v1

Mathematics

Functional Analysis

10 pages

Scientific paper

Theoretically speaking, there are four kinds of possibilities to define the random conjugate space of a random locally convex module. The purpose of this paper is to prove that among the four kinds there are only two which are universally suitable for the current development of the theory of random conjugate spaces: in this process we also obtain a somewhat surprising and crucial result that for a random normed module with base $(\Omega,{\cal F},P)$ such that $(\Omega,{\cal F},P)$ is nonatomic then the random normed module is a totally disconnected topological space when it is endowed with the locally $L^{0}-$convex topology.

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