  In this paper, we introduce the concepts of weaknorm, quasi-weaknorm on real
vector spaces. By these concepts, we introduce the concept of quasi-locally
convex topological vector spaces, which include locally convex topological
vector spaces as special cases. By the Fan-KKM theorem, we prove a fixed point
theorem in quasi-locally convex topological vector spaces, that is a natural
extension of Tychonoff fixed point theorem in locally convex topological vector
spaces. Then we provide an example to show that this extension is a proper
extension.
