Tools for infinite dimensional analysis

289 Pages·2020·3.273 MB·English

by  Jeremy J. Becnel

Tools for Infinite
Dimensional Analysis
Monographs and Research Notes in
Mathematics
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Summable Spaces and Their Duals, Matrix Transformations, and Geometric Properties
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Spectral Geometry of Partial Differential Operators (Open Access)
Michael Ruzhansky, Makhmud Sadybekov, Durvudkhan Suragan
Linear Groups: The Accent on Infinite Dimensionality
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Tools for Infinite Dimensional Analysis
Jeremy J. Becnel
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Tools for Infinite
Dimensional Analysis
Jeremy J. Becnel
Stephen F. Austin State University
First edition published 2021
by CRC Press
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Library of Congress Cataloging-in-Publication Data
Names: Becnel, Jeremy J., author.
Title: Tools for infinite dimensional analysis / Jeremy J. Becnel.
Description: First edition. | Boca Raton, FL : Chapman & Hall, CRC Press,
2021. | Series: Chapman & hall/crc monographs and research notes in
mathematics | Includes bibliographical references and index.
Identifiers: LCCN 2020037243 (print) | LCCN 2020037244 (ebook) | ISBN
9780367543662 (hardback) | ISBN 9781003088974 (ebook)
Subjects: LCSH: Dimensional analysis. | Linear topological spaces.
Classification: LCC QC20.7.D55 B43 2021 (print) | LCC QC20.7.D55
(ebook)
| DDC 515/.73--dc23
LC record available at https://lccn.loc.gov/2020037243
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ISBN: 9780367543662 (hbk)
ISBN: 9781003088974 (ebk)
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For Anna,
the light of my life.

Contents
Preface xi
1 Schwartz Space 1
1.1 The Schwartz Space . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Schwartz Norm Topology . . . . . . . . . . . . . . . . . 2
1.3 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 The multidimensional setting . . . . . . . . . . . . . . 13
1.4 Physical Context . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1 Hermite polynomials . . . . . . . . . . . . . . . . . . . 16
1.4.2 Creation and annihilation operators in L2(R,p(x)dx) . 20
1.5 Orthonormal Basis, Creation, and Annihilation Operators in
L2(R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5.1 Properties of the creation and annihilation operator . 25
1.6 Number Operator and the Spaces S (R) . . . . . . . . . . . . 32
p
1.7 The Projective Limit Topology on S(R) . . . . . . . . . . . . 38
1.8 L2 Norms on S(R) . . . . . . . . . . . . . . . . . . . . . . . . 40
1.9 Equivalence of the Topologies . . . . . . . . . . . . . . . . . 42
1.10 The Sequence Space . . . . . . . . . . . . . . . . . . . . . . . 47
1.11 Multidimensional . . . . . . . . . . . . . . . . . . . . . . . . 50
1.11.1 The operator approach. . . . . . . . . . . . . . . . . . 50
1.11.2 The L2 approach . . . . . . . . . . . . . . . . . . . . . 54
1.11.3 Topology on multidimensional Schwartz space . . . . . 54
1.11.4 Multidimensional sequence space . . . . . . . . . . . . 54
2 Nuclear Spaces 57
2.1 Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . 57
2.1.1 Special sets . . . . . . . . . . . . . . . . . . . . . . . . 64
2.1.2 Neighborhoods in topological vector spaces . . . . . . 68
2.1.3 Convergence and sequential spaces: A word of caution 70
2.1.3.1 Sequential spaces . . . . . . . . . . . . . . . 71
2.1.4 Seminorms . . . . . . . . . . . . . . . . . . . . . . . . 73
2.1.5 Topologies generated by families of topologies . . . . . 76
2.2 Countably-Normed Spaces . . . . . . . . . . . . . . . . . . . 79
2.2.1 Open sets in a countably-normed space . . . . . . . . 82
2.2.2 Metrizability . . . . . . . . . . . . . . . . . . . . . . . 83
2.2.3 Bounded sets in countably-normed spaces . . . . . . . 85
vii
viii Contents
2.2.4 Completeness in countably-normed spaces . . . . . . . 85
2.3 Nuclear Space Structure . . . . . . . . . . . . . . . . . . . . 87
2.3.1 Projective limit topology . . . . . . . . . . . . . . . . 92
2.3.2 Bounded sets in a nuclear space. . . . . . . . . . . . . 93
2.3.3 Compactness and the Heine–Borel property . . . . . . 96
2.3.4 Linear operators on nuclear spaces . . . . . . . . . . . 97
2.4 Nuclear Space Construction from a Hilbert Space and an Op-
erator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3 Dual Space 105
3.1 Linear Functionals . . . . . . . . . . . . . . . . . . . . . . . . 105
3.1.1 Linear functionals on nuclear spaces . . . . . . . . . . 108
3.2 Dual Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.2.1 Banach–Steinhaus . . . . . . . . . . . . . . . . . . . . 110
3.2.2 Hahn–Banach Theorems . . . . . . . . . . . . . . . . . 112
3.2.3 Weak topology . . . . . . . . . . . . . . . . . . . . . . 116
3.2.4 Banach–Alaoglu . . . . . . . . . . . . . . . . . . . . . 117
3.2.5 Strong topology . . . . . . . . . . . . . . . . . . . . . 121
3.3 The Dual of a Nuclear Space . . . . . . . . . . . . . . . . . . 124
3.4 Topologies on H(cid:48) . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.4.1 Weak topology on H(cid:48) . . . . . . . . . . . . . . . . . . 129
3.4.2 Strong topology on H(cid:48) . . . . . . . . . . . . . . . . . . 130
3.4.3 Gel’fand Triples and Rigged Hilbert Spaces . . . . . . 132
3.4.4 Strongly and weakly bounded sets in H(cid:48) . . . . . . . . 133
3.4.5 Weakly compact and strongly compact sets in H(cid:48) . . . 137
3.5 Reflexivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.5.1 Bounded sets in H revisited . . . . . . . . . . . . . . . 141
3.5.2 Weak convergence on H . . . . . . . . . . . . . . . . . 142
3.6 Convergence and Completeness in H(cid:48) . . . . . . . . . . . . . 143
3.6.1 Convergence of linear operators in H . . . . . . . . . . 149
3.7 Properties of the Topology on H(cid:48) and H . . . . . . . . . . 150
−p
3.8 Inductive Limit Topology . . . . . . . . . . . . . . . . . . . 152
3.8.1 Inductive limit topology on H(cid:48) . . . . . . . . . . . . . 154
3.8.2 Polars and uniform convergence . . . . . . . . . . . . . 155
3.9 Sequential Space . . . . . . . . . . . . . . . . . . . . . . . . 159
3.9.1 Sequential compactness . . . . . . . . . . . . . . . . . 161
3.10 First Countability (Lack Thereof) . . . . . . . . . . . . . . . 162
3.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
3.12 Abstract Kernel Theorem . . . . . . . . . . . . . . . . . . . . 165
3.13 White Noise Dual . . . . . . . . . . . . . . . . . . . . . . . . 167
3.13.1 The sequence space revisited . . . . . . . . . . . . . . 175
3.14 Schwartz Space Dual . . . . . . . . . . . . . . . . . . . . . . 179
3.15 Boxes and Continuous Functions . . . . . . . . . . . . . . . . 182
3.15.1 Continuous functions . . . . . . . . . . . . . . . . . . . 185
Contents ix
4 Probability Theory on Infinite Dimensional Spaces 189
4.1 Probability Theory Fundamentals . . . . . . . . . . . . . . . 189
4.1.1 Sigma algebras . . . . . . . . . . . . . . . . . . . . . . 189
4.1.1.1 Dynkin Class Theorem . . . . . . . . . . . . 190
4.1.2 Probability measures . . . . . . . . . . . . . . . . . . . 193
4.1.3 Outer measures . . . . . . . . . . . . . . . . . . . . . . 197
4.1.4 Riesz–Markov Theorem . . . . . . . . . . . . . . . . . 201
4.1.5 Characteristic functions and Bochner’s Theorem . . . 208
4.2 Product Sigma Algebras in Infinite Dimensions . . . . . . . . 214
4.3 Sigma Algebras on the Dual of a Nuclear Space . . . . . . . 215
4.4 Probability Measures on Infinite Dimensional Spaces . . . . . 217
4.4.1 Gaussian measure in infinite dimensions . . . . . . . . 222
4.4.2 Bochner-Minlos Theorem on sequence space . . . . . . 223
4.4.3 Bochner-Minlos Theorem . . . . . . . . . . . . . . . . 228
4.4.4 White noise space . . . . . . . . . . . . . . . . . . . . 230
Appendix: Banach and Hilbert Spaces 233
A Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 233
A.1 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 234
A.2 Span, Linear Dependence, and Dimension . . . . . . . 235
B Normed Linear Spaces . . . . . . . . . . . . . . . . . . . . . . 237
B.1 Closed Subspaces . . . . . . . . . . . . . . . . . . . . . 238
B.2 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . 239
C Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 239
C.1 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . 242
C.2 Orthonormal basis . . . . . . . . . . . . . . . . . . . . 244
C.3 Isometric Hilbert Spaces . . . . . . . . . . . . . . . . . 249
D Dual Space and Riesz Representation Theorem . . . . . . . . 250
E Operators on Hilbert Spaces . . . . . . . . . . . . . . . . . . 254
E.1 The adjoint A∗. . . . . . . . . . . . . . . . . . . . . . . 254
E.2 Closure and Essentially Self-adjoint Operators. . . . . 255
E.3 Bounded Linear Operators . . . . . . . . . . . . . . . 255
E.3.1 Hilbert–Schmidt Operators . . . . . . . . . . 258
Bibliography 263
Index 269