Functional Analysis: Vol. I

442 Pages·1996·9.117 MB·English

by  Yuri M. Berezansky, Zinovij G. Sheftel, Georgij F. Us (auth.)

Operator Theory
Advances and Applications
Vol. 85
Editor
I. Gohberg
Editorial Office: T. Kailath (Stanford)
School of Mathematical H.G. Kaper (Argonne)
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Functional Analysis
Vol. I
Y.M. Berezansky
Z.G. Shettel
G.F. Us
Translated from the Russian by Peter V. Malyshev
Birkhauser Verlag
Basel· Boston· Berlin
Authors' addresses:
Yurij M. Berezansky Georgij F. Us
Institute of Mathematics Mechanics and Mathematics Faculty
Ukrainian Academy of Sciences Kiev University
Repin str. 3 Vladimirskaya str. 64
252601 Kiev 252617 Kiev
Ukraine Ukraine
Zinovij G. Sheftel
Department of Mathematics
Pedagogical Institute
Sverdlov str. 53
250038 Chernigov
Ukraine
Originally published in 1990 by Vysha Shkola, Kiev.
1991 Mathematics Subject Classification 46-XX
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Berezanskij, Jurij M.:
Functional analysis / Y. M. Berezansky ; Z. G. Sheftel ; G. F.
Us. Trans!. from the Russian by Peter V. Malyshev. -Basel;
Boston; Berlin: Birkhauser
Einheitssacht.: Funkcional'nyj analiz <eng!.>
NE: Sefte!, Zinovij G.:; Us, Georgij F.:
Vo!. I (1996)
(Operator theory; Vo!' 85)
ISBN-I 3:978-3-0348-9939-0 e-ISBN-13:978-3-0348-9185-1
DOT: 10.1007/978-3-0348-9185-1
NE:GT
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the
rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and
storage in data banks. For any kind of use the permission of the copyright holder must be obtained.
© 1996 Birkhauser Verlag, P.O. Box 133, CHAOlO Basel, Switzerland
Printed on acid-free paper produced from chlorine-free pulp. TCF
00
Cover design: Heinz Hiltbrunner, Basel
ISBN 3-7643-5344-9
ISBN 0-8176-5344-9
Contents
Volume I
Introduction xvii
Chapter 1
Measure Theory 1
1 Operations on Sets. Ordered Sets ................................... 1
1.1 Operations on Sets n2 ......................................... 1
1.2 Ordered Sets. The Zorn Lemma ............................... 3
2 Systems of Sets .................................................... 4
2.1 Rings and Algebras of Sets .................................... 4
2.2 u-Rings and u-Algebras ....................................... 6
2.3 Generated Rings and Algebras ................................. 6
3 Measure of a Set. Simple Properties of Measures .................... 8
4 Outer Measure..................................................... 10
5 Measurable Sets. Extension of a Measure ........................... 13
6 Properties of Measures and Measurable Sets ........................ 19
7 Monotone Classes of Sets. Uniqueness of Extensions of Measures .... 24
8 Measures Taking Infinite Values .................................... 26
9 Lebesgue Measure of Bounded Linear Sets .......................... 28
10 Lebesgue Measure on the Real Line ................................. 35
11 Lebesgue Measure in the N-Dimensional Euclidean Space........... 40
12 Discrete Measures .................................................. 43
13 Some Properties of Nondecreasing Functions ........................ 44
13.1 Discontinuity Points of Monotone Functions ................... 44
13.2 Jump Function. Continuous Part of a
Nondecreasing Function....................................... 46
14 Construction of a Measure for a Given Nondecreasing Function.
Lebesgue-Stieltjes Measure ......................................... 48
15 Reconstruction of a Nondecreasing Function for a Given
Lebesgue-Stieltjes Measure ......................................... 53
16 Charges and Their Properties .................................... . . . 55
16.1 Concept of a Charge. Decomposition in Hahn's Sense.......... 55
16.2 Decomposition in Jordan's Sense .............................. 60
17 Relationship between Functions of Bounded Variation
and Charges ..................................................... . . . 62
vi CONTENTS VOLUME I
Chapter 2
Measurable Functions 67
1 Measurable Spaces. Measure Spaces. Measurable Functions .......... 68
2 Properties of Measurable Functions ................................. 71
3 Equivalence of Functions ........................................... 74
4 Sequences of Measurable Functions ................................. 76
5 Simple Functions. Approximation of Measurable Functions
by Simple Functions. The Luzin Theorem ........................... 85
Chapter 3
Theory of Integration 89
1 Integration of Simple Functions ..................................... 89
2 Integration of Measurable Bounded Functions ....................... 94
3 Relationship Between the Concepts of Riemann and
Lebesgue Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 Integration of Nonnegative Unbounded Functions ................... 103
5 Integration of Unbounded Functions with Alternating Sign. . . . . .. . . . 109
6 Limit Transition under the Sign of the Lebesgue Integral ............ 115
7 Integration over a Set of Infinite Measure ........................... 121
8 Summability and Improper Riemann Integrals ...................... 124
8.1 Integrals of Unbounded Functions ............................. 124
8.2 Integrals over Sets of Infinite Measure ......................... 125
9 Integration of Complex-Valued Functions ........................... 127
10 Integrals over Charges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 128
10.1 Integrals over Charges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
10.2 Integral over Complex-Valued Charges ......................... 128
11 Lebesgue-Stieltjes Integral and Its Relation to the
Riemann-Stieltjes Integral .......................................... 129
12 The Lebesgue Integral and the Theory of Series ..................... 131
CONTENTS VOLUME I vii
Chapter 4
Measures in the Products of Spaces.
Fubini Theorem........................ .......... . ...................... 133
1 Direct Product of Measurable Spaces. Sections of
Sets and Functions ................................................. 133
2 Product of Measures 136
3 The Fubini Theorem 139
4 Products of Finitely Many Measures 144
Chapter 5
Absolute Continuity and Singularity of Measures,
Charges, and Functions. Radon-Nikodym Theorem.
Change of Variables in the Lebesgue Integral............................ 147
1 Absolutely Continuous Measures and Charges....................... 147
2 Radon-Nikodym Theorem.......................................... 149
3 Radon-Nikodym Derivative. Change of Variables in the
Lebesgue Integral .................................................. 155
4 Mappings of Measure Spaces. Change of Variables in the
Lebesgue Integral. (Another Approach) ............................. 158
5 Singularity of Measures and Charges. Lebesgue Decomposition ...... 161
6 Absolutely Continuous Functions. Basic Properties .................. 164
7 Relationship Between Absolutely Continuous Functions
and Charges........................................................ 167
8 Newton-Leibniz Formula. Singular Functions. Lebesgue
Decomposition of a Function of Bounded Variation ................. 171
Chapter 6
Linear Normed Spaces and Hilbert Spaces 177
1 Topological Spaces ................................................. 177
2 Linear Topological Spaces .......................................... 179
3 Linear Normed and Banach Spaces ................................. 180
4 Completion of Linear Normed Spaces............................... 184
5 Pre-Hilbert and Hilbert Spaces ..................................... 188
6 Quasiscalar Product and Semi norms ................................ 192
viii CONTENTS VOLUME I
7 Examples of Banach and Hilbert Spaces ............................ 194
7.1 The Spaces eN and]RN ....................................... 194
7.2 The Space C(Q) .............................................. 195
7.3 The Space M(R) .............................................. 196
7.4 The Space Cm(O) ............................................. 197
7.5 The Space COO(O) ............................................ 197
8 Spaces of Summable Functions. Spaces Lp .......................... 198
8.1 Holder and Minkowski Inequalities. Definition
of the Spaces Lp .............................................. 198
8.2 Everywhere Dense Sets in Lp. Separability Conditions ......... 203
8.3 Different Types of Convergence in Lp .......................... 207
8.4 The Space lp .................................................. 208
8.5 The Space L (R, dp,) .......................................... 209
2
8.6 Essentially Bounded Functions. The Space Loo(R, dp,) .......... 209
8.7 The Space loo ................................................. 211
8.8 The Sobolev Spaces ........................................... 211
Chapter 7
Linear Continuous Functionals and Dual Spaces .......................... 215
1 Theorem on an Almost Orthogonal Vector. Finite
Dimensional Spaces ................................................ 215
2 Linear Continuous Functionals and Their Simple Properties.
Dual Space ......................................................... 219
3 Extension of Linear Continuous Functionals ......................... 223
3.1 Extension by Continuity ....................................... 223
3.2 Extension of a Functional Defined on a Subspace .............. 224
4 Corollaries of the Hahn-Banach Theorem ........................... 229
5 General Form of Linear Continuous Functionals in
Some Banach Spaces ............................................... 233
5.1 The Concept of a Schauder Basis .............................. 233
5.2 The Space Dual to lp (1 < p < 00) ............................. 234
5.3 The Space Dual to h .......................................... 236
5.4 The Space Dual to loo. Banach Limit .......................... 236
5.5 The Space Dual to Lp(R,dp,) (1 < p < 00) ....... ........ ..... 237
5.6 The Spaces Dual to Ll(R,dp,) and Loo(R,dp,) ............ ..... 240
5.7 The Space Dual to C(Q) ...................................... 240
6 Embedding of a Linear Normed Space in the Second Dual Space.
Reflexive Spaces .................................................... 244
CONTENTS VOLUME I ix
7 Banach-Steinhaus Theorem. Weak Convergence ..................... 246
7.1 Banach-Steinhaus Theorem.................................... 246
7.2 Weak Convergence of Linear Continuous Functionals ........... 247
7.3 Weak convergence in (C([a, b]))'. The Helly Theorems.......... 249
7.4 Weak Convergence in a Linear Normed Space.................. 251
8 Tikhonov Product. Weak Topology in the Dual Space ............... 254
8.1 Tikhonov Product of Topological Spaces....................... 254
8.2 Weak Topology in the Dual Space ............................. 255
9 Orthogonality and Orthogonal Projections in Hilbert Spaces.
General Form of a Linear Continuous Functional .................... 257
9.1 Orthogonality. Theorem on the Projection of a
Vector onto a Subspace........................................ 257
9.2 Orthogonal Sums of Subspaces ................................ 259
9.3 Linear Continuous Functionals in Hilbert Spaces ............... 261
10 Orthonormal Systems of Vectors and Orthonormal Bases in
Hilbert Spaces ...................................................... 262
10.1 Orthonormal Systems of Vectors. The Bessel Inequality........ 262
10.2 Orthonormal Bases in H. The Parseval Equality............... 264
10.3 Orthogonalization of a System of Vectors ...................... 266
10.4 Examples of Orthogonal Polynomials .......................... 267
10.5 Orthonormal Systems of Vectors of Arbitrary Cardinality 269
Chapter 8
Linear Continuous Operators 273
1 Linear Operators in Normed Spaces ................................ 273
2 The Space of Linear Continuous Operators ......................... 278
3 Product of Operators. The Inverse Operator ........................ 283
3.1 Product of Operators.......................................... 283
3.2 Normed Algebras .............................................. 284
3.3 The Inverse Operator ......................................... 285
4 The Adjoint Operator .............................................. 291
5 Linear Operators in Hilbert Spaces ................................. 296
5.1 Bilinear Forms ................................................ 296
5.2 Selfadjoint Operators .......................................... 298
5.3 Nonnegative Operators........................................ 299
5.4 Projection Operators .......................................... 300
5.5 Normal Operators ............................................. 301
5.6 Unitary and Isometric Operators .............................. 301
x CONTENTS VOLUME I
6 Matrix Representation of Operators in Hilbert Spaces ............... 304
6.1 Linear Operators in a Separable Space ......................... 304
6.2 Selfadjoint Operators .......................................... 306
6.3 Nonnegative Operators........................................ 306
6.4 Orthoprojectors ............................................... 307
6.5 Isometric Operators ........................................... 307
6.6 Jacobian Matrices ............................................. 308
7 Hilbert-Schmidt Operators ......................................... 309
7.1 Absolute Norm. . . . . .. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
7.2 Integral Hilbert-Schmidt Operators ............................ 312
8 Spectrum and Resolvent of a Linear Continuous Operator 315
Chapter 9
Compact Operators. Equations with Compact Operators 321
1 Definition and Properties of Compact Operators .................... 321
2 Riesz-Schauder Theory of Solvability of Equations with
Compact Operators ................................................ 327
3 Solvability of Fredholm Integral Equations .......................... 335
3.1 Some Classes of Integral Operators ............................ 335
3.2 Solvability of Fredholm Integral Equations of
the Second Kind .............................................. 337
3.3 Integral Equations with Degenerate Kernels ................... 339
4 Spectrum of a Compact Operator ................................... 342
5 Spectral Radius of an Operator ..................................... 346
5.1 Power Series with Operator Coefficients ....................... 346
5.2 Spectral Radius of a Linear Continuous Operator.............. 348
5.3 Method of Successive Approximations ......................... 349
6 Solution of Integral Equations of the Second Kind by the
Method of Successive Approximations .............................. 351
Chapter 10
Spectral Decomposition of Compact Selfadjoint Operators.
Analytic Functions of Operators ......................................... 355
1 Spectral Decomposition of a Compact Selfadjoint Operator ......... 355
1.1 One Property of Hermitian Bilinear Forms ..................... 355
1. 2 Theorem on Existence of an Eigenvector for a
Selfadjoint Compact Operator ................................. 356
1.3 Spectral Theorem for a Compact Selfadjoint Operator ......... 358