## Functional Analysis and Infinite-Dimensional GeometryThis book introduces the reader to the basic principles of functional analysis theory that are close to nonlinear analysis and topology. The presentation is self-contained, including many folklore results, and the proofs are accessible to students with the usual background in real analysis and topology. Several results are published here for the first time in a monograph. The text can be used in graduate courses or for independent study. It includes a large number of exercises of different levels of difficulty, accompanied by hints. The book is also directed to young researchers in functional analysis and can serve as a reference book, to areas of Banach space. |

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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 |

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### 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