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Introduction to Hilbert Space
Sterling K. Berberian
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AMS Chelsea Publishing
1961; 206 pp; hardcover
Volume: 287
ISBN-10: 0-8218-1912-7
ISBN-13: 978-0-8218-1912-8
List Price: US$34
Member Price: US$30.60
Order Code: CHEL/287.H
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See also:

Measure and Integration - Sterling K Berberian

From the Preface: "This textbook has evolved from a set of lecture notes ... In both the course and the book, I have in mind first- or second-year graduate students in Mathematics and related fields such as Physics ... It is necessary for the reader to have a foundation in advanced calculus which includes familiarity with: least upper bound (LUB) and greatest lower bound (GLB), the concept of function, \(\epsilon\)'s and their companion \(\delta\)'s, and basic properties of sequences of real and complex numbers (convergence, Cauchy's criterion, the Weierstrass-Bolzano theorem). It is not presupposed that the reader is acquainted with vector spaces ... , matrices ... , or determinants ... There are over four hundred exercises, most of them easy ... It is my hope that this book, aside from being an exposition of certain basic material on Hilbert space, may also serve as an introduction to other areas of functional analysis."

Reviews

"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

"Easy to read and, although the author had in mind graduate students, most of it is obviously appropriate for an advanced undergraduate course ... also a book which a reasonably good student might read on his own."

-- Mathematical Reviews

Table of Contents

Vector Spaces
  • 1. Complex vector spaces
  • 2. First properties of vector spaces
  • 3. Finite sums of vectors
  • 4. Linear combinations of vectors
  • 5. Linear subspaces, linear dependence
  • 6. Linear independence
  • 7. Basis, dimension
  • 8. Coda
Hilbert Spaces
  • 1. Pre-Hilbert spaces
  • 2. First properties of pre-Hilbert spaces
  • 3. The norm of a vector
  • 4. Metric spaces
  • 5. Metric notions in pre-Hilbert space; Hilbert spaces
  • 6. Orthogonal vectors, orthonormal vectors
  • 7. Infinite sums in Hilbert space
  • 8. Total sets, separable Hilbert spaces, orthonormal bases
  • 9. Isomorphic Hilbert spaces; classical Hilbert space
Closed Linear Subspaces
  • 1. Some notations from set theory
  • 2. Annihilators
  • 3. Closed linear subspaces
  • 4. Complete linear subspaces
  • 5. Convex sets, minimizing vector
  • 6. Orthogonal complement
  • 7. Mappings
  • 8. Projection
Continuous Linear Mappings
  • 1. Linear mappings
  • 2. Isomorphic vector spaces
  • 3. The vector space \(\scr{L}(\scr{V}, \scr{W})\)
  • 4. Composition of mappings
  • 5. The algebra \(\scr{L}(\scr{V})\)
  • 6. Continuous mappings
  • 7. Normed spaces, Banach spaces, continuous linear mappings
  • 8. The normed space \(\scr{L}_c(\scr{E}, \scr{F})\)
  • 9. The normed algebra \(\scr{L}_c(\scr{E})\), Banach algebras
  • 10. The dual space \(\scr{E}^{\prime}\)
Continuous Linear Forms in Hilbert Space
  • 1. Riesz-Frechet theorem
  • 2. Completion
  • 3. Bilinear mappings
  • 4. Bounded bilinear mappings
  • 5. Sesquilinear mappings
  • 6. Bounded sesquilinear mappings
  • 7. Bounded sesquilinear forms in Hilbert space
  • 8. Adjoints
Operators in Hilbert Space
  • 1. Manifesto
  • 2. Preliminaries
  • 3. An example
  • 4. Isometric operators
  • 5. Unitary operators
  • 6. Self-adjoint operators
  • 7. Projection operators
  • 8. Normal operators
  • 9. Invariant and reducing subspaces
Proper Values
  • 1. Proper vectors, proper values
  • 2. Proper subspaces
  • 3. Approximate proper values
Completely Continuous Operators
  • 1. Completely continuous operators
  • 2. An example
  • 3. Proper values of CC-operators
  • 4. Spectral theorem for a normal CC-operator
  • Appendix
  • Index
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