## Banach Space Theory: The Basis for Linear and Nonlinear AnalysisBanach spaces provide a framework for linear and nonlinear functional analysis, operator theory, abstract analysis, probability, optimization and other branches of mathematics. This book introduces the reader to linear functional analysis and to related parts of infinite-dimensional Banach space theory. Key Features: - Develops classical theory, including weak topologies, locally convex space, Schauder bases and compact operator theory - Covers Radon-Nikodým property, finite-dimensional spaces and local theory on tensor products - Contains sections on uniform homeomorphisms and non-linear theory, Rosenthal's L1 theorem, fixed points, and more - Includes information about further topics and directions of research and some open problems at the end of each chapter - Provides numerous exercises for practice The text is suitable for graduate courses or for independent study. Prerequisites include basic courses in calculus and linear. Researchers in functional analysis will also benefit for this book as it can serve as a reference book. |

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### Contents

2 HahnBanach and Banach Open Mapping Theorems | 53 |

3 Weak Topologies and Banach Spaces | 83 |

4 Schauder Bases | 179 |

5 Structure of Banach Spaces | 237 |

6 FiniteDimensional Spaces | 291 |

7 Optimization | 331 |

8 C1Smoothness in Separable Spaces | 383 |

9 Superreflexive Spaces | 429 |

13 Weakly Compactly Generated Spaces | 575 |

14 Topics in Weak Topologies on Banach Spaces | 617 |

15 Compact Operators on Banach Spaces | 657 |

16 Tensor Products | 687 |

17 Appendix | 733 |

751 | |

777 | |

781 | |

10 Higher Order Smoothness | 464 |

11 Dentability and Differentiability | 479 |

12 Basics in Nonlinear Geometric Analysis | 521 |

### Common terms and phrases

admits an equivalent Assume bounded operator canonical Cauchy closed convex closed subspace co(T compact operator compact set compact space Consider contains continuous function contradiction convergent convex set convex subset Corollary countable defined Definition denote dense dual norm dual pair Eberlein element equivalent norm Exercise exists extreme point f(xn finite Fréchet differentiable function f Gâteaux differentiable given hence Hilbert space Hint homeomorphic inequality infinite isometric isomorphic James boundary Lemma Let f linear functional Lipschitz locally convex space mapping Markushevich basis neighborhood nonempty normed space Note numbers one-to-one open set pointwise projection proof of Theorem Proposition prove quotient scalars Schauder basis semicontinuous separable Banach space sequence xn Show supremum topological space topological vector space unit ball w-compact weak topology