Introduction to Hilbert Space

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American Mathematical Soc., 1999 - Mathematics - 206 pages
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Completely 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

Closed linear subspaces
62
Orthogonal complement
70
Linear mappings
77
The vector space 0W
84
Continuous mappings
91
PROPER VALUES
163
Completely continuous operators
172
Appendix
189
Index
203
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