The purpose of this paper is to exhibit the relations between some basic results derived from the two kinds of topologies (namely the (ε, λ)-topology and the stronger locally L 0 -convex topology) for a random locally convex module. First, we give an extremely simple proof of the known Hahn-Banach extension theorem for L 0 -linear functions as well as its continuous variant. Then we give the relations between the hyperplane separation theorems in [D. Filipović, M. Kupper, N. Vogelpoth, Separation and duality in locally L 0 -convex modules, J. Funct. Anal. 256 (2009) 3996-4029] and a basic strict separation theorem in [T.X. Guo, H.X. Xiao, X.X. Chen, A basic strict separation theorem in random locally convex modules, Nonlinear Anal. 71 (2009) 3794-3804]: in the process we also obtain a very useful fact that a random locally convex module with the countable concatenation property must have the same completeness under the two topologies. As applications of the fact, we prove that most of the previously established principal results of random conjugate spaces of random normed modules under the (ε, λ)-topology are still valid under the locally L 0 -convex topology, which considerably enriches financial applications of random normed modules.

The purpose of this paper is to generalize the classical James theorem characterizing the reflexivity of Banach spaces to one characterizing the random reflexivity of complete random normed modules.

No abstract

Let (Ω, A, μ) be a probability space, K the scalar field R of real numbers or C of complex numbers,and (S, X ) a random normed space over K with base (Ω, A, μ). Denote the support of (S, X ) by E, namely E is the essential supremum of the set {A ∈ A : there exists an element p in S such that Xp(ω) > 0 for almost all ω in A}. In this paper, Banach-Alaoglu theorem in a random normed space is first established as follows: The random closed unit ball S * (1) = {f ∈ S * : X * f 1} of the random conjugate space (S * , X * ) of (S, X ) is compact under the random weak star topology on (S * , X * ) iff E A=: {E A | A ∈ A} is essentially purely μ-atomic (namely, there exists a disjoint family {An : n ∈ N } of at most countably many μ-atoms from E A such that E = ∞ n=1 An and for each element F in E A, there is an H in the σ-algebra generated by {An : n ∈ N } satisfying μ(F H) = 0), whose proof forces us to provide a key topological skill, and thus is much more involved than the corresponding classical case. Further, Banach-Bourbaki-Kakutani-Šmulian (briefly, BBKS) theorem in a complete random normed module is established as follows: If (S, X ) is a complete random normed module, then the random closed unit ball S(1) = {p ∈ S : Xp 1} of (S, X ) is compact under the random weak topology on (S, X ) iff both (S, X ) is random reflexive and E A is essentially purely μ-atomic. Our recent work shows that the famous classical James theorem still holds for an arbitrary complete random normed module, namely a complete random normed module is random reflexive iff the random norm of an arbitrary almost surely bounded random linear functional on it is attainable on its random closed unit ball, but this paper shows that the classical Banach-Alaoglu theorem and BBKS theorem do not hold universally for complete random normed modules unless they possess extremely simple stratification structure, namely their supports are essentially purely μ-atomic. Combining the James theorem and BBKS theorem in complete random normed modules leads directly to an interesting phenomenum: there exist many famous classical propositions that are mutually equivalent in the case of Banach spaces, some of which remain to be mutually equivalent in the context of arbitrary complete random normed modules, whereas the other of which are no longer equivalent to another in the context of arbitrary complete random normed modules unless the random normed modules in question possess extremely simple stratification structure. Such a phenomenum is, for the first time, discovered in the course of the development of random metric theory.Keywords: random normed module, random reflexivity, random weak star compactness, random weak compactness, stratification structure MSC(2000): 46A16, 46H25, 46B10, 54E45, 54E70

The purpose of this paper is to provide a random duality theory for the further development of the theory of random conjugate spaces for random normed modules. First, the complicated stratification structure of a module over the algebra L(µ, K) frequently makes our investigations into random duality theory considerably different from the corresponding ones into classical duality theory, thus in this paper we have to first begin in overcoming several substantial obstacles to the study of stratification structure on random locally convex modules. Then, we give the representation theorem of weakly continuous canonical module homomorphisms, the theorem of existence of random Mackey structure, and the random bipolar theorem with respect to a regular random duality pair together with some important random compatible invariants.Keywords: random duality, weakly continuous canonical module homomorphism, random compatible structure, random bipolar theorem MSC(2000): 46A20, 46A16, 46H05, 46H25