arXiv subjects

Guang Shi

Publications and source records attributed to Guang Shi.

39 records · Page 3Linked to original sources

arXiv2011

The Algebraic Structure of Finitely Generated $L^{0}(\mathcal{F},K)$-Modules and the Helly Theorem in Random Normed Modules

Let $K$ be the scalar field of real numbers or complex numbers and $L^{0}(\mathcal{F},K)$ the algebra of equivalence classes of $K-$valued random variables defined on a probability space $(Ω,\mathcal{F},P)$. In this paper, we first characterize the algebraic structure of finitely generated $L^{0}(\mathcal{F},K)$-modules and then combining the recently developed separation theorem in random locally convex modules we prove the Helly theorem in random normed modules with the countable concatenation property under the framework of random conjugate spaces at the same time a simple counterexample shows that it is necessary to require the countable concatenation property. By the way,we also give an application to the existence problem of the random solution of a system of random linear functional equations.

math.FA↗

arXiv2011

A geometric form of the Hahn-Banach extension theorem for L0 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 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 $(ε,λ)-$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.

math.FA↗

arXiv2010

The Goldstine-Weston theorem in random normed modules

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 $(ε,λ)$ 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.

math.FA↗