Discussion Of Related Property Of Some Topological Spaces

In this paper, we draw the definition of sub-normed2Z-linear space from the existing concept normed2-linear space and sub-normed Z-linear space, and research about some theorems of the traditional normed linear space would be still set up in sub-normed2Z-linear space. On the other hand, we research fixed point and best approximation in convex metric space. And study convexity, smoothness and norm differentiability in Banach space as well. There are three chapters in the paper.In Chapter1, we introduce the definition sub-normed2Z-linear space. Then researching sequence convergence, cauchy sequence and space convergence property. And on this basis, we study the Hahn-Banach theorem in sub-normed2Z-inner product linear space.In Chapter2, by using some properties of convex metric spaces, we show nec-essary conditions for the existence of fixed point and approximate fixed point of nonexpansive and quasi nonexpansive maps in a convex metric spaces. We obtain results on best approximation as a fixed point in a strictly convex metric spaces.In Chapter3, we discuss the convexity, smoothness and norm differentiability in Banach space. And obtain some new lemmas, then give new proofs for some existing results.