Principles of Advanced Mathematical Physics
A first consequence of this difference in texture concerns the attitude we must take toward some (or perhaps most) investigations in "applied mathe matics," at least when the mathematics is applied to physics. Namely, those investigations have to be rega
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W. Beiglbock
M. Goldhaber E. H. Lieb W. Thirring Series Editors
Robert D. Richtmyer
Principles of Advanced Mathematical Physics Volume I
[I]
Springer-Verlag New York
Heidelberg
Berlin
Robert D. Richtmyer Department of Physics and Astrophysics University of Colorado Boulder, Colorado 80309 USA Editors:
Wolf Beiglbock
Maurice Goldhaber
Institut fUr Angewandte Mathematik Universitat Heidelberg 1m Neuenheimer Feld 5 D-6900 Heidelberg I Federal Republic of Germany
Department of Physics Brookhaven National Laboratory Associated Universities, Inc. Upton, NY 11973 USA
Elliott H. Lieb
Walter Thirring
Department of Physics Joseph Henry Laboratories Princeton University PO. Box 708 Princeton, NJ 08540 USA
Institut fUr Theoretische Physik der Universitat Wien Boltzmanngasse 5 A-1090 Wien Austria
With 45 Figures
ISBN-13: 978-3-642-46380-8 DOl: 10.1007/978-3-642-46378-5
e-ISBN-13: 978-3-642-46378-5
Library of Congress Cataloging in Publication Data Richtmyer, Robert D. Principles of advanced mathematical physics. (Texts and monographs in physics) CONTENTS: v. 1. Hilbert and Banach spaces, distributions, operators, probability, applications to quantum mechanics, equations of evolution in physics. Includes index. I. Mathematical physics. I. Title. QC20. R56 530.1' 5 78-16494 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag.
© 1978 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1978
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Contents
xi
Preface
1
Hilbert Spaces 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 l.l 0 1.11
2
1
Review of pertinent facts about matrices and finitedimensional spaces I Linear spaces; normed linear spaces 3 Hilbert space: axioms and elementary consequences 4 Examples of Hilbert spaces 6 Cardinal numbers; separability; dimension 8 Orthonormal sequences II Subspaces; the projection theorem 14 Linear functionals; the Riesz-Fn!chet representation theorem Strong and weak convergence 16 Hilbert spaces of analytic functions 17 Polarization 17
19
Distributions; General Properties 2.1 2.2 2.3 2.4 2.5
Origin of the distribution concept 19 Classes of test functions; functions of type C g' 21 Notations for distributions; the bilinear form 22 The formal definition; the continuity of the functionals Examples of distributions 26
16
24 v
vi
Contents 2.6 Distributions as limits of sequences of functions; convergence of distributions 29 2.7 Differentiation and integration 31 2.8 Change of independent variable; symmetries 33 2.9 Restrictions, limitations, and warnings 35 2.10 Regularization 39 Appendix: A discontinuous linear functional 40
3
local Properties of Distributions
3.1 3.2 3.3 3.4 3.5 3.6
4
5
5.9 5.10 5.11 5.12 5.13
52
The space!/ 52 Tempered distributions 53 Growth at infinity 54 Fourier transformation in!/ 55 Fourier transforms of tempered distributions 56 The power spectrum 60
U Spaces 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8
6
Quick review of open and closed sets in IRn 43 Local properties def