The Goldstine-Weston theorem in random normed modules

TL;DR

This paper extends the classical Goldstine-Weston theorem to random normed modules, demonstrating the density of the natural embedding in the double conjugate space under various topologies.

Contribution

It generalizes the Goldstine-Weston theorem from normed spaces to the setting of random normed modules, incorporating the $( ext{epsilon}, ext{lambda})$ weak star and locally $L^{0}$-convex weak star topologies.

Findings

The image of a random normed module under the natural embedding is dense in its double conjugate space in the $( ext{epsilon}, ext{lambda})$ weak star topology.

The embedding is also dense in the double conjugate space with respect to the locally $L^{0}$-convex weak star topology if the module has the countable concatenation property.

Abstract

This article generalize the classical Goldstine-Weston theorem on normed spaces to one on random normed modules: the image of a random normed module $(E,\|\cdot\|)$ under the random natural embedding $J$ is dense in its double random conjugate space $E^{**}$ with respect to the $(\epsilon,\lambda)$ weak star topology; and $J(E)$ is also dense in $E^{**}$ with respect to the locally $L^{0}$-convex weak star topology if $E$ has the countable concatenation property.