An Introduction to Hilbert Space

by N. Young

• ISBN13: 9780521337175
• ISBN10: 0521337178
• Format: Paperback
• Copyright: 1988-07-29
• Publisher: Cambridge University Press

Summary

This textbook is an introduction to the theory of Hilbert spaces and its applications. The notion of a Hilbert space is a central idea in functional analysis and can be used in numerous branches of pure and applied mathematics. Dr. Young stresses these applications particularly for the solution of partial differential equations in mathematical physics and to the approximation of functions in complex analysis. Some basic familiarity with real analysis, linear algebra and metric spaces is assumed, but otherwise the book is self-contained. The book is based on courses given at the University of Glasgow and contains numerous examples and exercises (many with solutions). The book will make an excellent first course in Hilbert space theory at either undergraduate or graduate level and will also be of interest to electrical engineers and physicists, particularly those involved in control theory and filter design.

Table of Contents

Foreword	ix
Introduction	1	(3)
Inner product spaces	4	(9)
Inner product spaces as metric spaces	6	(5)
Problems	11	(2)
Normed spaces	13	(8)
Closed linear subspaces	15	(3)
Problems	18	(3)
Hilbert and Banach spaces	21	(10)
The space L2(a, b)	23	(3)
The Closest point property	26	(2)
Problems	28	(3)
Orthogonal expansions	31	(14)
Bessel's inequality	34	(1)
Pointwise and L2 convergence	35	(1)
Complete orthonormal sequences	36	(3)
Orthogonal complements	39	(3)
Problems	42	(3)
Classical Fourier series	45	(14)
The Fejer kernel	46	(6)
Fejer's theorem	52	(2)
Parseval's formula	54	(1)
Weierstrass' approximation theorem	54	(1)
Problems	55	(4)
Dual spaces	59	(8)
The Riesz-Frechet theorem	62	(2)
Problems	64	(3)
Linear operators	67	(22)
The Banach space G(E, F)	71	(1)
Inverses of operators	72	(3)
Adjoint operators	75	(3)
Hermitian operators	78	(2)
The spectrum	80	(2)
Infinite matrices	82	(1)
Problems	83	(6)
Compact operators	89	(16)
Hilbert-Schmidt operators	92	(4)
The spectral theorem for compact Hermitian operators	96	(6)
Problems	102	(3)
Sturm-Liouville systems	105	(14)
Small oscillations of a hanging chain	105	(6)
Eigenfunctions and eigenvalues	111	(3)
Orthogonality of eigenfunctions	114	(1)
Problems	115	(4)
Green's functions	119	(12)
Compactness of the inverse of a Sturm-Liouville operator	124	(4)
Problems	128	(3)
Eigenfunction expansions	131	(10)
Solution of the hanging chain problem	134	(4)
Problems	138	(3)
Positive operators and contractions	141	(16)
Operator matrices	144	(2)
Mobius transformations	146	(3)
Completing matrix contractions	149	(3)
Problems	152	(5)
Hardy spaces	157	(20)
Poisson's kernel	161	(3)
Fatou's theorem	164	(5)
Zero sets of H2 functions	169	(2)
Multiplication operators and infinite Toeplitz and Hankel matrices	171	(3)
Problems	174	(3)
Interlude: complex analysis and operators in engineering	177	(10)
Approximation by analytic functions	187	(16)
The Nehari problem	189	(1)
Hankel operators	190	(6)
Solution of Nehari's problem	196	(4)
Problems	200	(3)
Approximation by meromorphic functions	203	(18)
The singular values of an operator	204	(2)
Schmidt pairs and singular vectors	206	(4)
The Adamyan-Arov-Krein theorem	210	(9)
Problems	219	(2)
Appendix: square roots of positive operators	221	(4)
References	225	(1)
Answers to selected problems	226	(4)
Afterword	230	(6)
Index of notation	236	(2)
Subject index	238