A geometric form of the Hahn-Banach extension theorem for L0 linear functions and the Goldstine-Weston theorem in random normed modules

TL;DR

This paper develops a geometric version of the Hahn-Banach theorem for $L^{0}$-linear functions, proves its equivalence to the analytic form, and applies it to establish the Goldstine-Weston theorem in random normed modules under different topologies.

Contribution

It introduces a geometric form of the Hahn-Banach theorem for $L^{0}$-linear functions and applies it to prove the Goldstine-Weston theorem in random normed modules.

Findings

Geometric form of Hahn-Banach theorem is equivalent to the analytic form.

New proof of a strict separation theorem in random locally convex modules.

Goldstine-Weston theorem holds under $(\\epsilon,\lambda)$-topology and locally $L^{0}$-convex topology, with a counterexample for the latter.

Abstract

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 analytic form of the Hahn-Banach extension theorem. Further, we use the geometric form to give a new 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 $(\epsilon,\lambda)-$topology and the locally $L^{0}-$convex topology, and also provide a counterexample showing that the Goldstine-Weston theorem under the locally $L^{0}-$convex topology can only hold for random normed modules with the countable concatenation property.