An Introduction to Hilbert Space

Copertina anteriore
Cambridge University Press, 21 lug 1988 - 239 pagine
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.
 

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Sommario

Inner product spaces
4
11 Inner product spaces as metric spaces
6
12 Problems
11
Normed spaces
13
21 Closed linear subspaces
15
22 Problems
18
Hilbert and Banach spaces
21
31 The space L²a b
23
93 Orthogonality of eigenfunctions
114
94 Problems
115
Greens functions
119
101 Compactness of the inverse of a SturmLiouville operator
124
102 Problems
128
Eigenfunction expansions
131
111 Solution of the hanging chain problem
134
112 Problems
138

32 The closest point property
26
33 Problems
28
Orthogonal expansions
31
41 Bessels inequality
34
42 Pointwise and L² convergence
35
43 Complete orthonormal sequences
36
44 Orthogonal complements
39
45 Problems
42
Classical Fourier series
45
51 The Fejér kernel
46
52 Fejérs theorem
52
53 Parsevals formula
54
55 Problems
55
Dual spaces
59
61 The RieszFrechet theorem
62
62 Problems
64
Linear operators
67
71 The Banadi space LE F
71
72 Inverses of operators
72
73 Adjoint operators
75
74 Hermitian operators
78
75 The spectrum
80
76 Infinite matrices
82
77 Problems
83
Compact operators
89
81 Hilbert Schmidt operators
92
82 The spectral theorem for compact Hermitian operators
96
83 Problems
102
SturmLiouville systems
105
92 Eigenfunctions and eigenvalues
111
Positive operators and contractions
141
121 Operator matrices
144
122 Möbius transformations
146
123 Completing matrix contractions
149
124 Problems
152
Hardy spaces
157
131 Poissons kernel
161
132 Fatous theorem
164
133 Zero sets of H² functions
169
134 Multiplication operators and infinite Toeplitz and Hankel matrices
171
135 Problems
174
Interlude complex analysis and operators in engineering
177
Approximation by analytic functions
187
151 The Nehari problem
189
152 Hankel operators
190
153 Solution of Neharis problem
196
154 Problems
200
Approximation by meromorphic functions
203
161 The singular values of an operator
204
162 Schmidt pairs and singular vectors
206
163 The AdamyanArovKrein theorem
210
164 Problems
219
square roots of positive operators
221
References
225
Answers to selected problems
226
Afterword
230
Index of notation
236
Subject index
238
Copyright

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An Introduction to Hilbert Space
N. Young
Anteprima limitata - 1988
An Introduction to Hilbert Space
N. Young
Anteprima limitata - 1988

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a₁ algebra analytic functions Banach space best approximation bounded linear operator Chapter closed subspace compact Hermitian operator compact operator complete orthonormal sequence complex numbers contraction converges Corollary corresponding Deduce defined Definition Let denote diagonal differential equation e₁ eigenfunctions eigenvalue of RSL eigenvalues eigenvectors engineering example exists finite-dimensional follows Fourier series function f functional analysis H₁ Hence Hermitian operator Hilbert space Hilbert-Schmidt inequality inner product space integral operator invertible K₂ L-norm L²(a Lemma Let H linear functional linear span mathematicians mathematics maximizing vector non-zero normed space operator norm orthogonal orthonormal basis polynomial positive operators Problem Proof Prove rational function respect right hand side satisfies scalar multiplication Show singular value singular vector solution space and let space H spectral theorem Sturm-Liouville system Suppose supremum norm Theorem Let unique vector space xe H

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Informazioni bibliografiche

TitoloAn Introduction to Hilbert Space
Cambridge mathematical textbooks
AutoreN. Young
Edizioneillustrata, ristampa
EditoreCambridge University Press, 1988
ISBN0521337178, 9780521337175
Lunghezza239 pagine
  
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