Assuming only a basic knowledge of functional analysis, the book gives the reader a self-contained overview of the ideas and techniques in the development of modern Banach space theory. Special emphasis is placed on the study of the classical Lebesgue spaces L_p (and their sequence space analogues) and spaces of continuous functions. The authors also stress the use of bases and basic sequences techniques as a tool for understanding the isomorphic structure of Banach spaces. The aim of this text is to provide the reader with the necessary technical tools and background to reach the frontiers of research without the introduction of too many extraneous concepts. Detailed and accessible proofs are included, as are a variety of exercises and problems.

Written for: Graduate mathematics students, functional analysts

• Bases and Basic Sequences
• The Classical Sequence Spaces
• Special Types of Bases
• Banach Spaces of Continuous Functions
• L1(µ)-Spaces and C(K)-Spaces
• The L_p -Spaces for 1 =< p < ∞
• Factorization Theory
• Absolutely Summing Operators
• Perfectly Homogeneous Bases and Their Applications
• l_p-Subspaces of Banach Spaces
• Finite Representability of l_p -Spaces
• An Introduction to Local Theory
• Important Examples of Banach Spaces

• A Fundamental Notions
• B Elementary Hilbert Space Theory
• C Main Features of Finite-Dimensional Spaces
• D Cornerstone Theorems of Functional Analysis
• E Convex Sets and Extreme Points
• F The Weak Topologies
• G Weak Compactness of Sets and Operators