Topics in Banach Space Theory
This book provides a sequel treatise on classical and modern Banach space theory. It is mainly focused on the study of classical Lebesgue spaces Lp, sequence spaces lp, and Banach spaces of continuous functions. The early chapters use bases and basic sequences as a tool for understanding the isomorphic structure of Banach spaces. The next few chapters deal with C(K)-spaces (including Miljutin's theorem) and L1(μ)-spaces. A chapter discussing the basic theory of Lp-spaces includes notions of type and cotype and the next one presents the Maurey-Nikishin factorization theory. This leads to the Grothendieck theor y of absolutely summing operators. Other topics treated include perfectly homogeneous bases, the Ramsey theory, Rosenthal's l1 theorem, Tsirelson space, finite representability of lp spaces and an introduction to the local theory of Banach spaces (the John ellipsoid, Dvoretzky's theorem and the complemented subspace problem). The final chapter covers important examples of Banach spaces (and also a generalization of James space and constructions of spaces via trees).
Each chapter ends with many exercises and problems of varying difficulty giving further applications and extensions of the theory. There is a comprehensive bibliography (225 items). The book is understandable and requires only a basic knowledge of functional analysis (all the prerequisites assumed in the book are collected without proofs in the appendices). It can be warmly recommended to a broad spectrum of readers - to graduate students, young researchers and also to specialists in the field.