A First Course in Functional Analysis

257 Pages·2017·2.071 MB·English

by  Orr Moshe Shalit

A FIRST COURSE IN
FUNCTIONAL
ANALYSIS

A FIRST COURSE IN
FUNCTIONAL
ANALYSIS
ORR MOSHE SHALIT
Technion - Israel Institute of Technology
Haifa, Israel
CRC Press
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Library of Congress Cataloging‑in‑Publication Data
Names: Shalit, Orr Moshe.
Title: A first course in functional analysis / Orr Moshe Shalit.
Description: Boca Raton : CRC Press, [2016] | Includes bibliographical references and
index.
Identifiers: LCCN 2016045930| ISBN 9781498771610 (hardback : alk. paper) | ISBN
9781315367132 (ebook) | ISBN 9781498771627 (ebook) | ISBN  9781498771641 (ebook)
| ISBN 9781315319933 (ebook)
Subjects: LCSH: Functional analysis--Textbooks.
Classification: LCC QA320 .S45927 2016 | DDC 515/.7--dc23
LC record available at https://lccn.loc.gov/2016045930
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To my mother and my father, Malka and Meir Shalit

Contents
Preface xi
1 Introduction and the Stone-Weierstrass theorem 1
1.1 Background and motivation . . . . . . . . . . . . . . . . . . . 1
1.2 The Weierstrass approximation theorem . . . . . . . . . . . . 3
1.3 The Stone-Weierstrass theorem . . . . . . . . . . . . . . . . . 5
1.4 The Stone-Weierstrass theorem over the complex numbers . 8
1.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 10
2 Hilbert spaces 13
2.1 Background and motivation . . . . . . . . . . . . . . . . . . . 13
2.2 The basic definitions . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 The space of Lebesgue square integrable functions . . . . . . 21
2.5 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 25
3 Orthogonality, projections, and bases 29
3.1 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Orthogonal projection and orthogonaldecomposition . . . . 30
3.3 Orthonormal bases . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Dimension and isomorphism . . . . . . . . . . . . . . . . . . 38
3.5 The Gram-Schmidt process . . . . . . . . . . . . . . . . . . . 40
3.6 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 40
4 Fourier series 45
4.1 Fourier series in L2 . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Pointwise convergence of Fourier series (Dirichlet’s theorem) 50
4.3 Fej´er’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 *Proof of Dirichlet’s theorem . . . . . . . . . . . . . . . . . . 54
4.5 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 55
vii
viii Contents
5 Bounded linear operators on Hilbert space 61
5.1 Bounded operators . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Linear functionals and the Riesz representation theorem . . . 64
5.3 *The Dirichlet problem for the Laplace operator . . . . . . . 65
5.4 The adjoint of a bounded operator . . . . . . . . . . . . . . . 70
5.5 Special classes of operators . . . . . . . . . . . . . . . . . . . 72
5.6 Matrix representation of operators . . . . . . . . . . . . . . . 73
5.7 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 75
6 Hilbert function spaces 83
6.1 Reproducing kernel Hilbert spaces . . . . . . . . . . . . . . . 83
6.2 *The Bergman space . . . . . . . . . . . . . . . . . . . . . . 87
6.3 *Additional topics in Hilbert function space theory . . . . . 91
6.4 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 100
7 Banach spaces 103
7.1 Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 Bounded operators . . . . . . . . . . . . . . . . . . . . . . . . 108
7.3 The dual space . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.4 *Topological vector spaces . . . . . . . . . . . . . . . . . . . 113
7.5 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 114
8 The algebra of bounded operators on a Banach space 119
8.1 The algebra of bounded operators . . . . . . . . . . . . . . . 119
8.2 An application to ergodic theory . . . . . . . . . . . . . . . . 120
8.3 Invertible operators and inverses . . . . . . . . . . . . . . . . 124
8.4 *Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.5 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 131
9 Compact operators 137
9.1 Compact operators . . . . . . . . . . . . . . . . . . . . . . . 137
9.2 The spectrum of a compact operator . . . . . . . . . . . . . 139
9.3 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 142
10 Compact operators on Hilbert space 145
10.1 Finite rank operators on Hilbert space . . . . . . . . . . . . . 145
10.2 The spectral theorem for compact self-adjoint operators . . . 146
10.3 The spectral theorem for compact normal operators . . . . . 150
10.4 The functional calculus for compact normal operators . . . . 152
10.5 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 155
Contents ix
11 Applications of the theory of compact operators 161
11.1 Integral equations . . . . . . . . . . . . . . . . . . . . . . . . 161
11.2 *Functional equations . . . . . . . . . . . . . . . . . . . . . . 171
11.3 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 174
12 The Fourier transform 177
12.1 The spaces Lp(R), p [1, ) . . . . . . . . . . . . . . . . . . 177
∈ ∞
12.2 The Fourier transform on L1(R) . . . . . . . . . . . . . . . . 182
12.3 The Fourier transform on L2(R) . . . . . . . . . . . . . . . . 189
12.4 *Shannon’s sampling theorem . . . . . . . . . . . . . . . . . 193
12.5 *The multivariate Fourier transforms . . . . . . . . . . . . . 194
12.6 Additional exercises . . . . . . . . . . . . . . . . . . . . . . . 197
13 *The Hahn-Banach theorems 201
13.1 The Hahn-Banach theorems . . . . . . . . . . . . . . . . . . 201
13.2 The dual space, the double dual, and duality . . . . . . . . . 209
13.3 Quotient spaces . . . . . . . . . . . . . . . . . . . . . . . . . 211
13.4 Additional excercises . . . . . . . . . . . . . . . . . . . . . . 214
Appendix A Metric and topological spaces 217
A.1 Metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
A.2 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
A.3 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . 225
A.4 The Arzel`a-Ascoli theorem . . . . . . . . . . . . . . . . . . . 230
Symbol Description 233
Bibliography 235
Index 237