Introduction to Hilbert Space

by Berberian, Sterling K.

• ISBN13: 9780821819128
• ISBN10: 0821819127
• Format: Hardcover
• Copyright: 1999-03-01
• Publisher: Chelsea Pub Co

Summary

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.''

Table of Contents

Vector Spaces

Complex vector spaces

First properties of vector spaces

Finite sums of vectors

Linear combinations of vectors

Linear subspaces, linear dependence

Linear independence

Basis, dimension

Coda Hilbert Spaces

Pre-Hilbert spaces

First properties of pre-Hilbert spaces

The norm of a vector

Metric spaces

Metric notions in pre-Hilbert space; Hilbert spaces

Orthogonal vectors, orthonormal vectors

Infinite sums in Hilbert space

Total sets, separable Hilbert spaces, orthonormal bases

Isomorphic Hilbert spaces; classical Hilbert space Closed Linear Subspaces

Some notations from set theory

Annihilators

Closed linear subspaces

Complete linear subspaces

Convex sets, minimizing vector

Orthogonal complement

Mappings

Projection Continuous Linear Mappings

Linear mappings

Isomorphic vector spaces

The vector space $\scr{L}(\scr{V}, \scr{W})$

Composition of mappings

The algebra $\scr{L}(\scr{V})$

Continuous mappings

Normed spaces, Banach spaces, continuous linear mappings

The normed space $\scr{L}_c(\scr{E}, \scr{F})$

The normed algebra $\scr{L}_c(\scr{E})$, Banach algebras

The dual space $\scr{E}^{\prime}$ Continuous Linear Forms in Hilbert Space

Riesz-Frechet theorem

Completion

Bilinear mappings

Bounded bilinear mappings

Sesquilinear mappings

Bounded sesquilinear mappings

Bounded sesquilinear forms in Hilbert space

Adjoints Operators in Hilbert Space

Manifesto

Preliminaries

An example

Isometric operators

Unitary operators

Self-adjoint operators

Projection operators

Normal operators

Invariant and reducing subspaces Proper Values

Proper vectors, proper values

Proper subspaces

Approximate proper values Completely Continuous Operators

Completely continuous operators

An example

Proper values of CC-operators

Spectral theorem for a normal CC-operator

Appendix

Index

Table of Contents provided by Publisher. All Rights Reserved.