## Introduction to the Analysis of Normed Linear SpacesThis text is ideal for a basic course in functional analysis for senior undergraduate and beginning postgraduate students. John Giles provides insight into basic abstract analysis, which is now the contextual language of much modern mathematics. Although it is assumed that the student has familiarity with elementary real and complex analysis, linear algebra, and the analysis of metric spaces, the book does not assume a knowledge of integration theory or general topology. Its central theme concerns structural properties of normed linear spaces in general, especially associated with dual spaces and continuous linear operators on normed linear spaces. Giles illustrates the general theory with a great variety of example spaces. |

### What people are saying - Write a review

User Review - Flag as inappropriate

basic properties normed lines space

### Contents

NORMED LINEAR SPACE STRUCTURE AND EXAMPLES | 1 |

Exercises | 19 |

2 Classes of example spaces | 24 |

Exercises | 45 |

3 Orthonormal sets in inner product spaces | 50 |

Exercises | 64 |

SPACES OF CONTINUOUS LINEAR MAPPINGS | 67 |

Exercises | 86 |

13 Adjoint operators on Hilbert space | 181 |

Exercises | 193 |

14 Projection operators | 196 |

Exercises | 204 |

15 Compact operators | 206 |

Exercises | 214 |

Chapter VI SPECTRAL THEORY | 217 |

Exercises | 221 |

5 The shape of the dual | 92 |

Exercises | 110 |

THE EXISTENCE OF CONTINUOUS LINEAR FUNCTIONALS | 113 |

Exercises | 120 |

7 The natural embedding and reflexivity | 122 |

Exercises | 130 |

8 Subreflexivity | 132 |

Exercises | 138 |

Chapter IV THE FUNDAMENTAL MAPPING THEOREMS FOR BANACH SPACES | 139 |

Exercises | 150 |

10 The Open Mapping and Closed Graph Theorems | 153 |

Exercises | 158 |

11 The Uniform Boundedness Theorem | 163 |

Exercises | 169 |

Chapter V TYPES OF CONTINUOUS LINEAR MAPPINGS | 171 |

Exercises | 179 |

17 The spectrum of a continuous linear operator | 222 |

Exercises | 227 |

18 The spectrum of a compact operator | 230 |

Exercises | 235 |

19 The Spectral Theorem for compact normal operators on Hilbert space | 237 |

Exercises | 241 |

20 The Spectral Theorem for compact operators on Hilbert space | 243 |

Exercises | 250 |

APPENDIX | 252 |

A2 Numerical equivalence | 254 |

A3 Hamel basis | 256 |

Historical notes | 258 |

List of symbols | 269 |

272 | |

### Common terms and phrases

Baire space Banach space bounded characterisation closed graph closed linear subspace closed unit ball compact operator complex Hilbert space complex numbers continuous functions continuous linear functional continuous linear mapping continuous linear operator Corollary countable deduce defined Definition dense dimensional normed linear eigenvalues element Euclidean Example exists f(ek forallxeX functional f generalisation Given a normed Hahn-Banach Theorem Hamel basis Hilbert space hyperplane II.II implies inequality INI2 inner product space integral isometrically isomorphic k=l k=l ker f ll.ll ll.lloo metric space norm ll.ll normal operator normed linear space one.to.one Open Mapping Theorem orthogonal projection positive operator projection operator Proof proper closed linear properties Prove real numbers reflexive Remark scalar Schauder basis self.adjoint operators sequence xn space H Spectral Theorem spectrum subset theory topological isomorphism unique x e H x,y e