The Fixed Point Property of Quasi-Point-Separable Topological Vector Spaces

TL;DR

This paper introduces quasi-point-separable topological vector spaces, a broad class including locally convex spaces, and proves they have the fixed point property, extending Tychonoff's theorem.

Contribution

The paper defines quasi-point-separable spaces, shows they include many known classes, and proves they possess the fixed point property, broadening fixed point theory.

Findings

Quasi-point-separable spaces include locally convex and pseudonorm adjoint spaces.

Every quasi-point-separable Hausdorff space has the fixed point property.

The main theorem extends Tychonoff's fixed point theorem to a larger class of spaces.

Abstract

In this paper, we introduce the concept of quasi-point-separable topological vector spaces, which has the following important properties: 1.In general, the conditions for a topological vector space to be quasi-point-separable is not very difficult to check; 2.The class of quasi-point-separable topological vector spaces is very large that includes locally convex topological vector spaces and pseudonorm adjoint topological vector spaces as special cases; 3.Every quasi-point-separable Housdorrf topological vector space has the fixed point property (that is, every continuous self-mapping on any given nonempty closed and convex subset has a fixed point), which is the result of the main theorem of this paper (Theorem 4.1). Furthermore, we provide some concrete examples of quasi-point-separable topological vector spaces, which are not locally convex. It follows that the main theorem of this paper is a proper extension of Tychonoffs fixed point theorem on locally convex topological vector spaces.