On the random conjugate spaces of a random locally convex module

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: if the base (Ω, F, P) of a random normed module is nonatomic then the random normed module is a totally disconnected topological space when it is endowed with the locally L ⁰-convex topology.