Extreme Points, Exposed Points, Differentiability Points In CL-Spaces

The study of convexity of Banach spaces has played an important part in Banach spaces. Speaking of classification of Banach spaces by convexity, we should mention here that there are two extreme classes: One is the class of uniformly convex Banach spaces and the other is that of flat spaces. It is well-known that the former is a very useful class of Banach spaces . Results of theoretical research and their applications have spread a wide variaty of topics in both linear and nonlinear functional analysis. The study of various properties of CL-spaces has also brought mathematicians great attention (See, for instance, [5] , [11], [12], [13], [19]). The aim of this paper devotes to study a property of geometric and topological nature of Gateaux differentiability and Frechet differentiability points of the flat spaces.A real or complex Banach space is said to be a CL-space if its unit ball is the absolutely convex hull of every maximal convex subset of the unit sphere. If the unit ball is the closed absolutely convex hull of every maximal convex subset of the unit sphere, we say that the space is an almost CL-spae.This paper presents a property of geometric and topological nature of Gateaux differentiability points and Frechet differentiability points of almost CL-spaces. More precisely, if we denote by M a maximal convex set of the unit sphere of a CL-space X, and by C_M the cone generated by M, then all Gateaux differentiability points of X are just∪n-s(C_M), and all Frechet differentiability points of X are∪int(C_M), (where n-s(C_M) denotes the non-support points set of C_M).