首先给出关于随机共轭空间表示的一个深刻定理及其若干重要应用。然后，作为表示定理的应用，证明了如下基本定理：设B*为Banach空间B的共轭空间，（Ω,σ,υ）为任一概率空间，那么任一定义在（Ω,σ,υ）上取值于B*的弱*-υ-可测函数都弱*-等价于一个定义在（Ω,σ,υ）上取值于B*υ-可测函数的充要条件为B*关于（Ω,σ,υ）具有Radon-Nikodym性质

利用随机共轭空间的表示定理证明了一个一般的对偶表示定理。作为应用，统一并改进了关于Lebesgue-Bochner函数空间的对偶表示方面许多熟知的重要定理。

The aim of this note is to fill in a gap in our previous paper in this journal. Precisely, we give a new proof of the following theorem: let (Ω, A, υ) be a σ-finite measure space with υ(Ω)>0, 0<p<+, and Ya separable subspace of a Banach space X. Then Y is proximinal in X iff Lp(υ, X)

The central purpose of this paper is to prove the following theorem: let (Ω,σ υ) be a complete probability space, (B, Ⅱ Ⅱ) a normed linear space over the scalar field K, E: Ω-2B a separable random domain with linear subspace values, and f: GrE-K a continuous random linear operator, where GrE={(ω×lq Bx E(ω)} denotes the graph of E. Then there exists a continuous random lin-ear operator f2 Ω×B-K such that f(ω, x)=f(ω, x) A ωΩ, x E(ω), and sup{f(ω, x)Ⅱx B, IIxII≤I}=sup{If(ω, x)Ⅱx E(ω), IIxII≤1}, for each to in 11. For the case where E is not separable, a result similar to the above-stated theorem is also given, which generalizes and improves many previous results on random generalizations of the Hahn-Banach Theorem.

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.

首先对完备随机内积模上的几乎处处有界的随机线性泛函建立了Riesz表示定理，该定理不仅表明每个完备随机内积模都是随机自共轭的而且也改进了文[1]的主要结果；然后，作为Riesz表示定理的应用，还证明了如下基本定理：设（Ω,συ）为任一概率空间，（H（·，·））为任一不可分的Hilbert空间，那么任一弱随机元V：Ω-H日必弱等价于某个强可测的随机元V：Ω-H

In this paper, a characterization of continuous module homomorphisms on random seminormed modules is first given; then the characterization is further used to show that the Hahn-Banach type of extension theorem is still true for continuous module homomorphisms on random semi-normed modules.