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

In recent thirty years, the results about the Drop property, Ekeland variational principle, Differentiation Theory and the Asplund property are very important and very good on locally convex spaces. But the research of locally convex spaces about convexity and smoothness is relative poor. In this paper, several K convexity and K smoothness in locally convex spaces are studied. We discussed their characteristic, properties and the relative relation. We divide this paper into five chapters. Chapter 1: The knowledge of preparation. Chapter 2: In this section, K-uniform convexity and K-uniform smoothness in locally convex spaces are introduced. It is showed some characteristic of locally convex K-uniform convexity and locally convex K-uniform smoothness, and obtained the dual theorem between K-uniform convexity and K-uniform smoothness. The definitions of K-uniform convexity and K-uniform smoothness in locally convex spaces not only the generalization of uniform convexity and uniform smoothness in locally convex spaces, but also the generalization of K-uniform convexity and K-uniform smoothness in Banach spaces. In the case of K=1,the results in this section included all results of uniform convexity and uniform smoothness in locally convex spaces. Chapter 3: In this section, K-strong convexity and K-strong smoothness in locally convex spaces are introduced. It is showed some characteristic of locally convex K-strong convexity and locally convex K-strong smoothness, and obtained the dual theorem between K-strong convexity and K-strong smoothness. The definitions of K-strong convexity and K-strong smoothness in locally convex spaces not only the generalization of strong convexity and strong smoothness in locally convex spaces, but also the generalization of K-strong convexity and K-strong smoothness in Banach spaces. In the case of K=1,the results in this section included all results of strong convexity and strong smoothness in locally convex spaces. Chapter 4: In this section, the K-strict convexity and K-smoothness in locally convex spaces are studied. As a result of that, some characterizations of K-strict convexity and K-smoothness are obtained. This work helps us to understand the characteristic of K-strict convexity and K-smoothness in locally convex spaces from different sides. We obtained the dual theorem between K-strict convexity and K-smoothness. Chapter 5: In this section, the relations among K-uniform convexity(K-uniform smoothness), K-strong convexity(K-strong smoothness), K-strict convexity(K-strict smoothness) are discussed.