## Nonarchimedean Functional AnalysisThis book grew out of a course which I gave during the winter term 1997/98 at the Universitat Munster. The course covered the material which here is presented in the first three chapters. The fourth more advanced chapter was added to give the reader a rather complete tour through all the important aspects of the theory of locally convex vector spaces over nonarchimedean fields. There is one serious restriction, though, which seemed inevitable to me in the interest of a clear presentation. In its deeper aspects the theory depends very much on the field being spherically complete or not. To give a drastic example, if the field is not spherically complete then there exist nonzero locally convex vector spaces which do not have a single nonzero continuous linear form. Although much progress has been made to overcome this problem a really nice and complete theory which to a large extent is analogous to classical functional analysis can only exist over spherically complete field8. I therefore allowed myself to restrict to this case whenever a conceptual clarity resulted. Although I hope that thi8 text will also be useful to the experts as a reference my own motivation for giving that course and writing this book was different. I had the reader in mind who wants to use locally convex vector spaces in the applications and needs a text to quickly gra8p this theory. |

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### Contents

Chapter I Foundations | 1 |

Nonarchimedean Fields | 2 |

Seminorms | 6 |

Normed Vector Spaces | 8 |

Locally Convex Vector Spaces | 13 |

Constructions and Examples | 19 |

Spaces of Continuous Linear Maps | 27 |

Completeness | 35 |

Admissible Topologies | 83 |

Reflexivity | 86 |

Compact Limits | 89 |

Chapter IV Nuclear Maps and Spaces | 101 |

Topological Tensor Products | 102 |

Completely Continuous Maps | 113 |

Nuclear Spaces | 119 |

Nuclear Maps | 125 |

Frechet Spaces | 45 |

The Dual Space | 50 |

Chapter II The Structure of Banach Spaces | 59 |

Nonreflexivity | 64 |

Chapter III Duality Theory | 67 |

cCompact and Compactoid Submodules | 68 |

Polarity | 76 |

Traces | 136 |

Fredholm Theory | 140 |

151 | |

Notations | 153 |

155 | |

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### Common terms and phrases

A C V According to Lemma According to Prop assume B C V Banach space barrelled bijection bornological bounded and c-compact bounded o-submodule bounded open lattice bounded subset canonical map Cauchy net closure compact map compactoid continuous linear form continuous linear map continuous seminorm converges convex A'-vector spaces convex vector spaces Corollary denote duality map equicontinuous family of lattices family of seminorms final topology finite rank operator follows from Prop Frechet space hand side Hausdorff hence implies injective K-vector Let L C V lim Vn locally convex A'-vector locally convex topology locally convex vector map f nonarchimedean normed vector space o-module open lattice L C V pL(v projective tensor product Proposition pseudo-polar quasi-complete quotient reflexive Remark resp scalar seminorm q spherically complete subspace topology Suppose surjective tensor product topology topological isomorphism vector subspace weak topology zero vector

### Popular passages

Page 152 - Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan 19 (1967), 366-383.

Page 152 - WH Schikhof. Locally convex spaces over non-spherically complete valued fields I, II. Bull. Soc. Math. Belg.

Page 151 - GRUSON, L., Theorie de Fredholm p-adique, Bull. Soc. Math. France 94, 67-95 (1966). [4] GRUSON, L., Cathégories d'espaces de Banach ul tramé triques, Bull.

Page 152 - Une notion de compacite dans la theorie des espaces vectoriels topologiques, Indagationes Math. 27. 182-189 (1965) van der Put M., Reflexive non-archimedean Banach spaces, Indagationes Math.

### References to this book

P-adic Geometry: Lectures from the 2007 Arizona Winter School Matthew Baker,David Savitt,Dinesh S. Thakur No preview available - 2008 |