Geometrical Local Structures of Banach Spaces and Dvoretzky Theorem

Abstract

This study is mainly devoted to the local theory of Banach spaces. In this theory, one obtains information regarding an in nite dimensional Banach space from its local structure - the collection of all its nite dimensional subspaces or quotients. The rst major result of the local theory is Dvoretzky's Theorem 1960. In his original approach, Dvoretzky used the Haar measure on the Grassmann manifold of all kdimensional subspaces of Rn: The objective of the rst three chapters is to present self-contained proofs of two fundamental results Dvorezky- Rogers theorem and Dvoretzky theorem on almost spherical (rather ellipsoidal) sections of convex bodies. Then we display consequences of Dvoretzky theorem in the fourth chapter, and one of its consequences is a stronger version of Dvoretzky-Rogers theorem. Finally, we get new results in this direction in the last chapter. So, this M. Sc. thesis is organized as follows: In chapter ] 1, we study Haar measure, the quotient topology and how the subspace topology on the Stiefel manifold gives a quotient topology on the Grassmann manifold. Also, we mention preliminaries on Banach spaces and linear operators and study Banach-Mazur distance. The most important concept studied was the correspondence between Banach spaces and symmetric convex bodies. This allows one to use arguments from convex geometry in functional analysis and vice versa. Furthermore, we studied the geometric notion of Banach-Mazur