An Extension of Tychonoffs Fixed Point Theorem to Quasi-point Separable Topological Vector Spaces

TL;DR

This paper extends Tychonoff's fixed point theorem to quasi-point separable topological vector spaces using generalized m-quasiconvex functionals, broadening its applicability beyond locally convex spaces.

Contribution

The paper introduces the concept of quasi-point separable spaces and extends fixed point theorems to these spaces, generalizing previous results on pseudonorm adjoint spaces.

Findings

Every pseudonorm adjoint space is quasi-point separable.

A fixed point theorem is proved for quasi-point separable spaces using the Fan-KKM theorem.

The extension includes locally convex spaces as special cases.

Abstract

In this paper, we introduce the concepts of m-quasiconvex, originally m-quasiconvex,and generalized m-quasiconvex functionals on topological vector spaces. Then we extend the concept of point separable topological vector spaces (by the topological dual spaces) to quasi-point separable topological vector spaces by families of generalized m-quasiconvex functionals. We show that every pseudonorm adjoint topological vector space, which includes locally convex topological vector spaces as special cases, is quasi-point separable. By the Fan-KKM theorem, we prove a fixed point theorem on quasi-point separable topological vector spaces. It is an extension of the fixed point theorem on pseudonorm adjoint topological vector spaces proved in [8]. Therefore, it is a properextension of Tychonoffs fixed point theorem on locally convex topological vector spaces, which are demonstrated by some examples.