Study On The K Convexity And K Smoothness In Locally Convex Spaces

Let X be a real linear spaces, P the family of separated seminorms, (X,P) a dual, (X,T_p) a locally convex space generated by P. In this paper, we spead some geometry theories on k convexity, k smoothness in Banach spaces and convexity, smoothness in dual (X,P). And introduce k-uniformly convex, locally k-uniformly convex , weakly locally k-uniformly convex , k-strongly convex, k-extremely convex , k-strict convex and corresponding k-uniformly smooth, locally k-uniformly smooth,weakly locally k-uniformly smooth,k-strongly smooth,k-extremely smooth, k-smooth of dual (X,P). We also discuss the relationship and dual property of them.