The Study On Three Kinds Of K-Convexity And K-Smoothness In Locally Convex Spaces

Let X be a real linear spaces, Pa family of separated seminorms on X ,(X,T_P) a locallyconvex space generated by P,(X,P) a dual. This thesis deals mainly with three aspects of convexity (smoothness):The first section, the notions of property (WM) and (WM)~* for dual (X,P) are introduced the raletionship between some convexity (smoothness) are discussed, four equivalent theorems are established, and the dual property between the property (WM) and (WM )~* are proved. Then, theequivalent notion of k-strongly smooth and k -very smooth for dual(X,P) are gived, furthermore,the result that k -strongly smooth for dual (X, P) is a strict generalization of corresponding notionfor Banach space is obtained.In the second section ,we introduce the notions of k -uniformly extreme convex, k-uniformlyextreme smooth for dual(X,P), disscuss the relationship between them and the other k -convexity(k -smoothness), and explore the dual property between k -uniformly extreme convex and k -uniformly extreme smooth.Then, on condition P -reflexivity, we obtain further equivalence of the dual relationship between k -uniformly extreme convex and k -uniformly extreme smooth and the relationship between them and other k -convexity (k -smoothness).In the third section, we give the notions of (weakly) midpoint locally k -uniformly convex and(weakly) midpoint locally k -uniformly smooth for dual(X,P), research the relationship betweenthem and other k-convexity (k-smoothness) and a part of the dual property between (weakly) midpoint locally k -uniformly convex and (weakly) midpoint locally k -uniformly smooth.