L0-线性函数的Hahn-Banach扩张定理的几何形式与随机赋范模中的Goldstine-Weston定理

本文给出了关于L⁰- 线性函数的Hahn-Banach 扩张定理的几何形式并证明这个几何形式等价于它的代数形式. 进一步, 我们利用这个几何形式给出了随机局部凸模中熟知的基本分离定理的一个新的且简单的证明. 最后, 利用这个分离定理, 我们同时在两种拓扑 —(ε, λ)- 拓扑和局部L⁰- 凸拓扑下证明了随机赋范模中的Goldstine-Weston 稠密性定理, 并举出一个反例说明在局部L⁰- 凸拓扑下如果随机赋范模不具有可数连接性质, 则Goldstine-Weston 稠密性定理不一定成立.

A geometric form of the Hahn-Banach extension theorem for L^0-linear functions and the Goldstine-Weston theorem in random normed modules

In this paper, we present a geometric form of the Hahn-Banach extension theorem for L^0-linear functions and prove that the geometric form is equivalent to the algebraic form of the Hahn-Banach extension theorem. Further, we use the geometric form to give a new and simpler proof of a known basic strict separation theorem in random locally convex modules. Finally, using the basic strict separation theorem we establish the Goldstine-Weston theorem in random normed modules under the two kinds of topologies---the （ε ,λ）-topology and the locally L^0-convex topology, while we also provide a counterexample showing that the Goldstine-Weston theorem under the locally L^0-eonvex topology may not hold for random normed modules without the countable concatenation property.