This monograph is based on an earlier publication of P. Habala, P. Hájek and V. Zizler (Introduction to Banach Spaces I, II) published by Matfyzpress, Prague, in 1996. An enlarged team of authors have expanded the contents of these lecture notes, adding updated information in the geometric theory of Banach spaces.

The first three chapters on basic concepts in Banach spaces present the central material of the book: the Hahn-Banach theorem, the open mapping theorems and weak topologies disseminated by such refinements as the Krein-Milman theorem, James boundary, Ekeland’s variational principle and the Bishop-Phelps theorem, together with material on the Riesz-Schauder theory of compact operators. Chapter 4 deals with locally convex spaces (Mackey topologies, Choquet’s representation theorem and Kaplansky’s theorem). The structure of Banach spaces (projections, complementability and Schur’s property) and Schauder bases are discussed in Chapters 5 and 6. Chapters 8 and 9 are about the differentiability of norms (Šmulyan’s test, Fréchet differentiability of convex functions) and uniform convexity and uniform smoothness (also local reflexivity, superreflexivity, Enflo’s renorming). Smoothness and structure (variational principles, smooth approximation, Lipschitz homeomorphisms, etc.) form the subject of Chapter 9. The monograph contains a huge number of exercises, some for students’ drill and others is to extend the theory presented in the text.

This book can be warmly recommended to everyone interested in functional analysis, and Banach space theory in particular. It serves also as a textbook in courses for students in probability, physics, or engineering. Graduate students and researchers surely will find a lot of material from the field, as well as a source of inspiration.

Reviewer:

jl