文档介绍：Applications of
\'Iathernaiical Physics
.by
-So-bole'v
Volume Seven.
of 'lathomatical Nliiiraft IIs
Translations of Mathematical Monographs Volume 7
Applications of
FUNCTIONAL ANALYSIS
in Mathematical Physics
by
S. L. Sobolev
AMERICAN MATHEMATICAL SOCIETY
PROVIDENCE, RHODE ISLAND
1963
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Translated from the Russian by F. E. Browder
Publication aided by grant NSF-G12381
from the
NATIONAL SCIENCE FOUNDATION
Library of Congress Card Number 63-15658
Copyright © 1963 by the American Mathematical Society
All rights reserved. No portion of this book may be reproduced
without the written permission of the publisher.
Second Printing, 1964
Printed in the United States of America
CONTENTS
AUTHOR'S PREFACE Vii
CHAPTER I. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS
§1. INTRODUCTION 1
1. Summable functions (1). 2. The Holder and Minkowski inequalities (3).
3. The reverse of the Holder and Minkowski inequalities (7).
§2. BASIC PROPERTIES OF THE SPACES Lp 9
1. Norms. Definitions (9). 2. The Riesz-Fischer Theorem (11). 3. Con-
tinuity in the large of functions in Lp (11). 4. Countable s (13).
§3. LINEAR FUNCTIONALS ON LP 16
1. Definitions. Boundedness of linear functionals (16). 2. Clarkson's in-
equalities (17). 3. Theorem on the general form of linear functionals (22).
4. Convergence of functionals (25).
§4. COMPACTNESS OF SPACES 28
1. Definition pactness (28). 2. A theorem on pactness (29).
3. A theorem on pactness (30). 4. Proof of the theorem on strong
compactness (31).
§5. GENERALIZED DERIVATIVES 33
1. Basic definitions (33). 2. Derivatives of averaged functions (35). 3. Rules
for differentiation (37). 4. Independence of the domain (39).
§6. PROPERTIES OF INTEGRALS OF POTENTIAL TYPE 42
1. Integrals of potential typ