## Topics in Banach Space Theory, Volume 10This book grew out of a one-semester course given by the second author in 2001 and a subsequent two-semester course in 2004-2005, both at the Univ- sity of Missouri-Columbia. The text is intended for a graduate student who has already had a basic introduction to functional analysis; the aim is to give a reasonably brief and self-contained introduction to classical Banach space theory. Banach space theory has advanced dramatically in the last 50 years and webelievethatthetechniquesthathavebeendevelopedareverypowerfuland should be widely disseminated amongst analysts in general and not restricted to a small group of specialists. Therefore we hope that this book will also prove of interest to an audience who may not wish to pursue research in this area but still would like to understand what is known about the structure of the classical spaces. Classical Banach space theory developed as an attempt to answer very natural questions on the structure of Banach spaces; many of these questions date back to the work of Banach and his school in Lvov. It enjoyed, perhaps, its golden period between 1950 and 1980, culminating in the de?nitive books by Lindenstrauss and Tzafriri [138] and [139], in 1977 and 1979 respectively. The subject is still very much alive but the reader will see that much of the basic groundwork was done in this period. |

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### Contents

Bases and Basic Sequences | 1 |

The Classical Sequence Spaces | 29 |

Special Types of Bases | 51 |

Banach Spaces of Continuous Functions | 73 |

The LpSpaces for 1 p | 125 |

Problems | 161 |

Absolutely Summing Operators | 195 |

Perfectly Homogeneous Bases and Their Applications | 221 |

An Introduction to Local Theory | 289 |

Important Examples of Banach Spaces | 309 |

A Fundamental Notions | 327 |

Main Features of FiniteDimensional Spaces | 335 |

E Convex Sets and Extreme Points | 341 |

G Weak Compactness of Sets and Operators | 347 |

References | 353 |

365 | |

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argument assume Banach space bases block basic sequence bounded called canonical basis chapter closed subspace complemented subspace complete consider constant construction contains continuous converges convex Corollary cotype deduce define Definition denote dense dual embedding embeds equivalent example exists extended fact factorization fails finite finitely representable function given gives Hence Hilbert space holds implies inequality infinite infinite-dimensional injective integers isometrically isomorphic Lemma Li(u linear measure metric natural norm normalized Note Notice numbers obtain operator particular pick positive probability problem projection Proof Proposition prove reflexive relatively Remark respect result satisfies scalars separable Show space X subsequence subset summing Suppose Theorem theory topology Tsirelson space unconditional basis unique unit vectors weak weakly compact yields