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 (epsilon,lambda)-topology and the locally L-0-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, L-q(S*) congruent to (L-p(S))', where p and q are a pair of Holder conjugate numbers with 1 <= p < +infinity, S a random normed module, S* the random conjugate space of S, L-p(S)(L-q(S*)) the corresponding L-p (resp., L-q) space derived from S (resp., S*), and (L-p(S))' the ordinary conjugate space of L-p(S).