  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.
