## 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. |

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### 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 |

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a-integral absolutely continuous accessible apply associated Banach space basis belongs bilinear form Bochner integrable Borel bounded Chapter choose closed coincide complete condition consider constant contains converges Corollary corresponding countable crossnorm defined definition denote dual space duality easy elements embedding equivalent example Exercise exists extension fact factorization finite dimensional subspace finite rank operator follows function Furthermore give given hence ideal identified implies inequality infimum injective tensor product Li(u linear mapping Loo(u Lp(u metric approximation property nuclear operators obtain operator p-summing pair positive measure projective norm projective tensor product Proof Proposition prove Radon-Nikodým property reader recall reflexive regular representation respect result satisfies scalar seen separable sequence Show subset summable suppose tensor norm tensor product Theorem uniform values vector measure weak weakly compact zero