Introduction to Tensor Products of Banach Spaces

Authors: Raymond A. Ryan



This book is intended as an introduction to the theory of tensor products of Banach spaces. The prerequisites for reading the book are a first course in Functional Analysis and in Measure Theory, as far as the Radon-Nikodym theorem. The book is entirely self-contained and two appendices give addi- tional material on Banach Spaces and Measure Theory that may be unfamil- iar to the beginner. No knowledge of tensor products is assumed. Our viewpoint is that tensor products are a natural and productive way to understand many of the themes of modern Banach space theory and that "tensorial thinking" yields insights into many otherwise mysterious phenom- ena. We hope to convince the reader of the validity of this belief. We begin in Chapter 1 with a treatment of the purely algebraic theory of tensor products of vector spaces. We emphasize the use of the tensor product as a linearizing tool and we explain the use of tensor products in the duality theory of spaces of operators in finite dimensions. The ideas developed here, though simple, are fundamental for the rest of the book.

Table of contents

Front Matter
Pages i-xiv

Tensor Products
Pages 1-13

The Projective Tensor Product
Pages 15-43

The Injective Tensor Product
Pages 45-69

The Approximation Property
Pages 71-92

The Radon-Nikodym Property
Pages 93-126

The Chevet-Saphar Tensor Products
Pages 127-158

Tensor Norms
Pages 159-185

Operator Ideals
Pages 187-199

Back Matter
Pages 201-225