## Introduction to Tensor Products of Banach SpacesThis 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. |

### Contents

I | 1 |

III | 5 |

IV | 7 |

V | 9 |

VI | 10 |

VII | 12 |

VIII | 15 |

X | 22 |

XXX | 103 |

XXXI | 108 |

XXXII | 114 |

XXXIII | 122 |

XXXIV | 125 |

XXXV | 127 |

XXXVII | 133 |

XXXVIII | 140 |

### Other editions - View all

### Common terms and phrases

2-summing a-integral a-nuclear absolutely continuous Banach ideal Banach space bilinear form Bochner integrable bounded bilinear form bounded linear functional bounded variation C(Bx canonical embedding canonical mapping closed subspace closed unit ball compact operator compact subset converges weakly Corollary defined denote dual space duality equivalent Exercise factorization finite dimensional subspace finite rank operator follows Furthermore Grothendieck hence Hilbert space Hilbertian inequality infimum injective norm injective tensor product integral bilinear form integral norm integral operator isometrically isomorphic l₁ L₁(µ linear mapping measurable sets metric approximation property nuclear norm nuclear operators numbers p-summing operators Pettis integrable Pietsch integral positive measure projective norm projective tensor product Proof Proposition quotient operator Rademacher functions Radon-Nikodým property reflexive regular Borel representation Schauder basis Show simple functions subspace suppose tensor norm Theorem uniform crossnorm values vector measure vector space XÔY