On Some Basic Theorems of Continuous Module Homomorphisms between Random Normed Modules

We first prove the resonance theorem, closed graph theorem, inverse operator theorem, and open mapping theorem for module homomorphisms between random normed modules by simultaneously considering the two kinds of topologies—the -topology and the locally -convex topology for random normed modules. Then, for the future development of the theory of module homomorphisms on complete random inner product modules, we give a proof with better readability of the known orthogonal decomposition theorem and Riesz representation theorem in complete random inner product modules under two kinds of topologies. Finally, to connect module homomorphism between random normed modules with linear operators between ordinary normed spaces, we give a proof with better readability of the known result connecting random conjugate spaces with classical conjugate spaces, namely, , where and are a pair of H？lder conjugate numbers with a random normed module, the random conjugate space of the corresponding (resp., ) space derived from (resp., ), and the ordinary conjugate space of 1. Introduction The theory of probabilistic metric spaces initiated by K. Menger and subsequently developed by Schweizer and Sklar begins the study of randomizing the traditional space theory of functional analysis, where the randomness of “distance” or “norm” is expressed by probability distribution functions; compare [1]. The original notions of random metric spaces and random normed spaces occur in the course of the development of probabilistic metric and normed spaces, whereas the random distance between two points in a random metric space or the random norm of a vector in a random normed space is described by nonnegative random variables on a probability space; compare [1]. Probabilistic normed spaces are often endowed with the -topology and not locally convex in general; a serious obstacle to the deep development of probabilistic normed spaces is that the taditional theory of conjugate spaces does not universally apply to probabilistic normed spaces. Although the traditional theory of conjugate spaces does not universally apply to random normed spaces either, the additional measure-theoretic structure and the stronger geometric structure peculiar to a random normed space enable us to introduce the notion of an almost surely bounded random linear functional and establish its Hahn-Banach extension theorem, which leads to the idea of the theory of random conjugate spaces for random normed spaces; compare [2–4]. The further development of the theory of random conjugate spaces motivates us to present the