## Exercises in Functional AnalysisThe understanding of results and notions for a student in mathematics requires solving ex ercises. The exercises are also meant to test the reader's understanding of the text material, and to enhance the skill in doing calculations. This book is written with these three things in mind. It is a collection of more than 450 exercises in Functional Analysis, meant to help a student understand much better the basic facts which are usually presented in an introductory course in Functional Analysis. Another goal of this book is to help the reader to understand the richness of ideas and techniques which Functional Analysis offers, by providing various exercises, from different topics, from simple ones to, perhaps, more difficult ones. We also hope that some of the exercises herein can be of some help to the teacher of Functional Analysis as seminar tools, and to anyone who is interested in seeing some applications of Functional Analysis. To what extent we have managed to achieve these goals is for the reader to decide. |

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### Contents

Open closed and bounded sets in normed spaces | 3 |

11 Exercises | 4 |

12 Solutions | 11 |

Linear and continuous operators on normed spaces | 36 |

21 Exercises | 37 |

22 Solutions | 42 |

Linear and continuous functional Reflexive spaces | 68 |

32 Solutions | 72 |

Baires category The open mapping and closed graph theorems | 213 |

92 Solutions | 220 |

Part II Hilbert spaces | 241 |

Hilbert spaces general theory | 243 |

101 Exercises | 245 |

102 Solutions | 250 |

The projection in Hilbert spaces | 271 |

Linear and continuous operators on Hilbert spaces | 305 |

The distance between sets in Banach spaces | 86 |

42 Solutions | 92 |

Compactness in Banach spaces Compact operators | 107 |

51 Exercises | 108 |

52 Solutions | 115 |

The Uniform Boundedness Principle | 147 |

62 Solutions | 155 |

The HahnBanach theorem | 175 |

72 Solutions | 180 |

Applications for the HahnBanach theorem | 195 |

82 Solutions | 199 |

121 Exercises | 306 |

122 Solutions | 318 |

Part III General topological spaces | 366 |

Linear topological and locally convex spaces | 368 |

132 Solutions | 377 |

The weak topologies | 403 |

142 Solutions | 412 |

444 | |

List of Symbols | 447 |

449 | |

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### Common terms and phrases

A C X absolutely convergent an)neN G Baire category Banach space chapter closed linear subspace compact operator compact set consider continuous functional continuous operator contradiction convex set deduce defined denote dense fc=i finite finite-dimensional G l2 Hahn-Banach extension Hausdorff hence Hilbert space hypothesis inequality inner product isometry kerx Let H Let us suppose linear and continuous linear space linear subspace linearly independent n=l n=l n=l oo neighborhood norm convergent normed space obtain oo oo open set orthogonal projection orthonormal pointwise Prove reflexive space relatively compact scalar self-adjoint seminorm solution for exercise subset surjective theorem it follows topological space Uniform Boundedness Principle Vn G N Vx G X weak compact weak topology whence