Advanced Courses of Mathematical Analysis I

Preface

Contents

This is an introductory survey of the classification of Banach spaces as metric spaces, where the maps are (nonlinear) uniformly continuous maps or, more specifically, Lipschitz maps. We describe basic results which show that the uniform theory and the linear theory are different but that, nevertheless, some linear features of a Banach space are preserved under uniform homeomorphisms.

We present an introduction to the study of the local dual spaces of a Banach space. We describe with detail the main properties of this concept and give several characterizations. These characterizations allow us to show examples of local dual spaces for many classical spaces of sequences or functions.

The aim of this paper is to give a survey of results and ideas concerning orbits of operators and related notions of weak and polynomial orbits. These concepts are closely related to the invariant subspace/subset problem. Most of the proofs are not given in full details, we rather try to indicate the basic ideas. The central problems in the field are also formulated.

We review the concepts of Baire’s categories, porosity, and null sets in metric and Banach spaces, and then illustrate the generic approach to nonlinear problems by presenting and discussing several simple examples of its applications to nonexpansive mapping theory. An extension theorem for contractive mappings is also included.

Absolute-valued algebras are fully surveyed. Some attention is also paid to Banach spaces underlying complete absolute-valued algebras.