Geometry of Banach Spaces and Duality Mapping

Abstract

In this chapter, we are mainly concern with geometrical structures such as convexity and smoothness of Banach spaces. Indeed, various kind of convexity and smoothness of Banach spaces play an important role in the existence and approximation of fixed points of nonlinear mappings. The necessary concepts of the geometry of normed spaces—strict convexity and uniform convexity—are also discussed. This chapter also deals with useful properties of duality mappings that interplay with these geometrical structures of Banach spaces. In Sect. 2.1, we deal with strict convexity while Sect. 2.2 mainly concern with uniform convexity. In Sect. 2.3, we discuss modulus of convexity. In Sect. 2.4, we mainly concern with smoothness of Banach spaces. Section 2.5 mainly deals with the concept of duality mapping from a Banach space X to its dual $X^*$. A discussion on these is important for a better understanding of the properties of duality mapping.

The enchanting charms of this sublime science, mathematics reveal only to those who have the courage to go deeply into it.

Carl Friedrich Gauss

The study of convex sets is a branch of geometry, analysis and linear algebra that has numerous connections with other areas of mathematics and serves to unify many apparently diverse mathematical phenomena.

Victor Klee (1950)