## Introduction to Hilbert SpaceCompletely self-contained ... All proofs are given in full detail ... recommended for unassisted reading by beginners ... For teaching purposes this book is ideal. --Proceedings of the Edinburgh Mathematical Society The book is easy to read and, although the author had in mind graduate students, most of it is obviously appropriate for an advanced undergraduate course. It is also a book which a reasonably good student might read on his own. --Mathematical Reviews This textbook evolved from a set of course notes for first- or second-year graduate students in mathematics and related fields such as physics. It presents, in a self-contained way, various aspects of geometry and analysis of Hilbert spaces, including the spectral theorem for compact operators. Over 400 exercises provide examples and counter-examples for definitions and theorems in the book, as well as generalization of some material in the text. Aside from being an exposition of basic material on Hilbert space, this book may also serve as an introduction to other areas of functional analysis. The only prerequisite for understanding the material is a standard foundation in advanced calculus. The main notions of linear algebra, such as vector spaces, bases, etc., are explained in the first chapter of the book. |

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### Contents

VECTOR SPACES | 3 |

Linear combinations of vectors | 11 |

Linear independence | 17 |

Coda | 24 |

Metric spaces | 33 |

Metric notions in preHilbert space Hilbert spaces | 39 |

Inﬁnite sums in Hilbert space | 49 |

Isomorphic Hilbert spaces classical Hilbert space | 55 |

The normed ce 8fF | 100 |

The dual space 8 | 109 |

Bilinear mappings | 116 |

Sesquilinear mappings | 123 |

Bounded sesquilinear forms in Hilbert space | 130 |

OPERATORS IN HILBERT SPACE | 139 |

Unitary operators | 145 |

Projection operators | 151 |

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### Common terms and phrases

Akxk AP-value Banach algebra Banach space bijective called Cauchy sequence classical Hilbert space closed linear subspace closed subset complete complex numbers continuous linear form continuous linear mapping convergent deﬁned Deﬁnition denoted equivalent Example exists ﬁnite dimension ﬁnitely non-zero sequences ﬁrst ﬁxed following conditions functions on a,b hence Hilbert space SC hyponormal implies inﬁnite injective invariant invertible isometric Lemma linear combination metric space non-zero vector normed space Notation null space orthogonal orthonormal basis orthonormal sequence pre-Hilbert space proof of Theorem proper value proper vector Q SC real numbers scalar product self-adjoint operator sense of Deﬁnition sense of Exercise sequence of vectors sesquilinear form sesquilinear mappings set of vectors shown space of continuous space of ﬁnitely space of n-ples subspace of SC Suppose surjective Theorem I I Txly unitary space vector space isomorphism vector x Q x,y Q