The fixed point property of quasi-point-separable topological vector 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-pointseparable 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 Tychonoff’s fixed point theorem on locally convex topological vector spaces.