Table Of ContentTools for Infinite
Dimensional Analysis
Monographs and Research Notes in
Mathematics
Series Editors:
John A. Burns, Thomas J. Tucker, Miklos Bona, Michael Ruzhansky
About the Series
This series is designed to capture new developments and summarize what is known
over the entire field of mathematics, both pure and applied. It will include a broad
range of monographs and research notes on current and developing topics that will
appeal to academics, graduate students, and practitioners. Interdisciplinary books
appealing not only to the mathematical community but also to engineers, physicists,
and computer scientists are encouraged.
This series will maintain the highest editorial standards, publishing well-developed
monographs as well as research notes on new topics that are final, but not yet refined
into a formal monograph. The notes are meant to be a rapid means of publication for
current material where the style of exposition reflects a developing topic.
Summable Spaces and Their Duals, Matrix Transformations, and Geometric Properties
Feyzi Basar, Hemen Dutta
Spectral Geometry of Partial Differential Operators (Open Access)
Michael Ruzhansky, Makhmud Sadybekov, Durvudkhan Suragan
Linear Groups: The Accent on Infinite Dimensionality
Martyn Russel Dixon, Leonard A. Kurdachenko, Igor Yakov Subbotin
Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s,
Volume I
Yoshihiro Sawano, Giuseppe Di Fazio, Denny Ivanal Hakim
Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s,
Volume II
Yoshihiro Sawano, Giuseppe Di Fazio, Denny Ivanal Hakim
Tools for Infinite Dimensional Analysis
Jeremy J. Becnel
Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial
Differential Equations
Luca Lorenzi, Abdelaziz Rhandi
For more information about this series please visit: https://www.crcpress.com/Chapman--
HallCRC-Monographs-and-Research-Notes-in-Mathematics/book-series/CRCMONRESNOT
Tools for Infinite
Dimensional Analysis
Jeremy J. Becnel
Stephen F. Austin State University
First edition published 2021
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
© 2021 Jeremy J. Becnel
CRC Press is an imprint of Taylor & Francis Group, LLC
The right of Jeremy J. Becnel to be identified as author of this work has been asserted by him in ac-
cordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988.
Reasonable efforts have been made to publish reliable data and information, but the author and pub-
lisher cannot assume responsibility for the validity of all materials or the consequences of their use.
The authors and publishers have attempted to trace the copyright holders of all material reproduced
in this publication and apologize to copyright holders if permission to publish in this form has not
been obtained. If any copyright material has not been acknowledged please write and let us know so
we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information stor-
age or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, access www.copyright.
com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA
01923, 978-750-8400. For works that are not available on CCC please contact mpkbookspermis-
[email protected]
Trademark notice: Product or corporate names may be trademarks or registered trademarks and are
used only for identification and explanation without intent to infringe.
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
LC ebook record available at https://lccn.loc.gov/2020037244
ISBN: 9780367543662 (hbk)
ISBN: 9781003088974 (ebk)
Typeset in Computer Modern font
by KnowledgeWorks Global Ltd
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
