## Functional Analysis and Infinite-Dimensional GeometryBanach spaces provide a framework for linear and nonlinear functional analysis, operator theory, abstract analysis, probability, optimization, and other branches ofmathematics. This book is intended as an introduction to linear functional analysis and to some parts of infinite-dimensional Banach space theory. The first seven chapters are directed mainly to undergraduate and grad uate students. We have strived to make the text easily readable and as self-contained as possible. In particular, we proved many basic facts that are considered "folklore." An important part of the text is a large number of exercises with detailed hints for their solution. They complement the material in the chapters and contain many important results. The last fivechapters introduce the reader to selected topics in the theory of Banach spaces related to smoothness and topology.This part of the book isintended as an introduction to and a complement ofexisting books on the subject ([Bea], [BeLi], [DGZ3]' [Disl], [Dis2], [Fab], [JoL3], [LiT2], [Phe2]' [Woj]). Some material is presented here for the first time in a monograph form. For further reading in this area, we recommend for instance [Gil], [God4], [Gue], [JoL3], [Kec], [LjSo], [Neg], [MeNe]' [Oxt], [RoJa], [Sem], [Sin3], [TaI2], and [Yael. The text is based on graduate courses taught at the University ofAlberta in Edmonton in the years 1984-1997. These courses were also taken by many senior students in the Honors undergraduate program in Edmonton. |

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### Contents

Basic Concepts in Banach Spaces | 1 |

HahnBanach and Banach Open Mapping Theorems | 37 |

Weak Topologies | 63 |

Locally Convex Spaces | 107 |

Structure of Banach Spaces | 137 |

Schauder Bases | 161 |

Compact Operators on Banach Spaces | 203 |

Exercises | 231 |

Uniform Convexity | 285 |

Smoothness and Structure | 313 |

Weakly Compactly Generated Spaces | 357 |

Topics in Weak Topology | 387 |

Bx w Polish | 409 |

431 | |

445 | |

Differentiability of Norms | 241 |

### Common terms and phrases

admits an equivalent Assume basic sequence bounded linear operator canonical Cauchy choose closed convex closed subspace compact operator compact set compact space Consider contains contradiction convergent convex function convex set Corollary countable Define Definition denote dense set DGZ3 dual norm Eberlein compact equivalent norm exists extreme points Frechet differentiable function f G Bx G Sx Gateaux differentiable Given hence Hilbert space Hint homeomorphic inequality isometric isomorphic isomorphic copy James boundary Lemma Let f Let X,Y linear functional Lipschitz locally convex space Markushevich basis metric space metrizable neighborhood norm topology normed space Note numbers one-to-one open set previous exercise projection proof of Theorem Proposition prove satisfies scalars Schauder basis semicontinuous separable Banach space Show supremum T(Bx uniformly convex unit ball vector space weak topology weakly compact set X,Y be Banach