Analysis III

476 Pages·2009·8.45 MB·English

by  Herbert Amann

Herbert Amann
Joachim Escher
Analysis III
Translated from the German
by Silvio Levy and Matthew Cargo
Birkhäuser
Basel · Boston · Berlin
Authors:
Herbert Amann Joachim Escher
Institut für Mathematik Institut für Angewandte Mathematik
Universität Zürich Universität Hannover
Winterthurerstr. 190 Welfengarten 1
8057 Zürich 30167 Hannover
Switzerland Germany
e-mail: [email protected] e-mail: [email protected]
Originally published in German under the same title by Birkhäuser Verlag, Switzerland
© 2001 by Birkhäuser Verlag
2000 Mathematics Subject Classification: 28-01, 28A05, 28A20, 28A25, 28B05, 58-01,
58A05, 28A10, 58C35
Library of Congress Control Number: 2008939525
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.
ISBN 3-7643-7479-2 Birkhäuser Verlag, Basel – Boston – Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the
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Foreword
Thisthirdvolumeconcludesourintroductiontoanalysis,whereinwefinishlaying
the groundwork needed for further study of the subject.
As with the first two, this volume contains more material than can treated
in a single course. It is therefore important in preparing lectures to choose a
suitable subset of its content; the remainder can be treated in seminars or left
to independent study. For a quick overview of this content, consult the table of
contents and the chapter introductions.
Thisbookisalsosuitableasbackgroundforothercoursesorforselfstudy. We
hope thatits numerousglimpsesinto moreadvancedanalysiswillarousecuriosity
and so invite students to further explore the beauty and scope of this branch of
mathematics.
In writing this volume, we counted on the invaluable help of friends, col-
leagues, staff, and students. Special thanks go to Georg Prokert, Pavol Quittner,
Olivier Steiger, and Christoph Walker, who worked through the entire text crit-
ically and so helped us remove errors and make substantial improvements. Our
thanksalsogoesouttoCarlheinzKneiselandBeaWollenmann,wholikewiseread
the majority of the manuscript and pointed out various inconsistencies.
Withouttheinestimableeffortofour“typesettingperfectionist”,thisvolume
could not have reached its present form: her tirelessness and patience with TEX
andother softwarebroughtnotonly the end product, but alsonumerous previous
versions,toahighdegreeofperfection. Forthiscontribution,shehasourgreatest
thanks.
Finally, it is our pleasure to thank Thomas Hintermann and Birkha¨user for
their usual flexibility and friendly cooperation.
Zu¨rich and Hannover, July 2001 H. Amann and J. Escher
vi Foreword
Forewordto the English translation
We are again much obliged to Silvio Levy and Matt Cargo for their careful and
accurate translation of this last part of the original German treatise. Special
thanks go to Thomas Hempfling from Birkha¨user Verlag for rendering possible
this translation so that our analysis course is now available to a larger audience.
Zu¨rich and Hannover, January 2009 H. Amann and J. Escher
Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Chapter IX Elements of measure theory
1 Measurable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
σ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
The Borel σ-algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
The second countability axiom . . . . . . . . . . . . . . . . . . . . . . . 6
Generating the Borel σ-algebra with intervals . . . . . . . . . . . . . . 8
Bases of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . 9
The product topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Product Borel σ-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Measurability of sections . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Set functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Properties of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Null sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 Outer measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
The construction of outer measures . . . . . . . . . . . . . . . . . . . . 24
The Lebesgue outer measure . . . . . . . . . . . . . . . . . . . . . . . . 25
The Lebesgue–Stieltjes outer measure . . . . . . . . . . . . . . . . . . . 28
Hausdorff outer measures . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Measurable sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
The σ-algebra of μ∗-measurable sets. . . . . . . . . . . . . . . . . . . . 33
Lebesgue measure and Hausdorff measure . . . . . . . . . . . . . . . . 35
Metric measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 The Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
The Lebesgue measure space . . . . . . . . . . . . . . . . . . . . . . . . 40
The Lebesgue measure is regular . . . . . . . . . . . . . . . . . . . . . 41
viii Contents
A characterizationof Lebesgue measurability. . . . . . . . . . . . . . . 44
Images of Lebesgue measurable sets . . . . . . . . . . . . . . . . . . . . 44
The Lebesgue measure is translation invariant . . . . . . . . . . . . . . 47
A characterizationof Lebesgue measure. . . . . . . . . . . . . . . . . . 48
The Lebesgue measure is invariant under rigid motions . . . . . . . . . 50
The substitution rule for linear maps . . . . . . . . . . . . . . . . . . . 51
Sets without Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . 53
Chapter X Integration theory
1 Measurable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simple functions and measurable functions . . . . . . . . . . . . . . . . 62
A measurability criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 64
Measurable R-valued functions . . . . . . . . . . . . . . . . . . . . . . . 67
The lattice of measurable R-valued functions . . . . . . . . . . . . . . . 68
Pointwise limits of measurable functions . . . . . . . . . . . . . . . . . 73
Radon measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2 Integrable functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The integral of a simple function . . . . . . . . . . . . . . . . . . . . . 80
The L -seminorm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
1
The Bochner–Lebesgue integral . . . . . . . . . . . . . . . . . . . . . . 84
The completeness of L . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
1
Elementary properties of integrals . . . . . . . . . . . . . . . . . . . . . 88
Convergence in L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
1
3 Convergence theorems . . . . . . . . . . . . . . . . . . . . . . . . . . .
Integration of nonnegative R-valued functions . . . . . . . . . . . . . . 97
The monotone convergence theorem . . . . . . . . . . . . . . . . . . . . 100
Fatou’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Integration of R-valued functions . . . . . . . . . . . . . . . . . . . . . 103
Lebesgue’s dominated convergence theorem . . . . . . . . . . . . . . . 104
Parametrizedintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4 Lebesgue spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Essentially bounded functions . . . . . . . . . . . . . . . . . . . . . . . 110
The H¨older and Minkowski inequalities . . . . . . . . . . . . . . . . . . 111
Lebesgue spaces are complete . . . . . . . . . . . . . . . . . . . . . . . 114
L -spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
p
Continuous functions with compact support . . . . . . . . . . . . . . . 118
Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Continuous linear functionals on L . . . . . . . . . . . . . . . . . . . . 121
p
Contents ix
5 The n-dimensional Bochner–Lebesgue integral . . . . . . . . . . . . 128
Lebesgue measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 128
The Lebesgue integral of absolutely integrable functions . . . . . . . . 129
A characterizationof Riemann integrable functions . . . . . . . . . . . 132
6 Fubini’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Maps defined almost everywhere . . . . . . . . . . . . . . . . . . . . . . 137
Cavalieri’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Applications of Cavalieri’s principle . . . . . . . . . . . . . . . . . . . . 141
Tonelli’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Fubini’s theorem for scalar functions . . . . . . . . . . . . . . . . . . . 145
Fubini’s theorem for vector-valued functions . . . . . . . . . . . . . . . 148
Minkowski’s inequality for integrals . . . . . . . . . . . . . . . . . . . . 152
A characterizationof L (Rm+n,E) . . . . . . . . . . . . . . . . . . . . 157
p
A trace theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7 The convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Defining the convolution . . . . . . . . . . . . . . . . . . . . . . . . . . 162
The translation group. . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Elementary properties of the convolution . . . . . . . . . . . . . . . . . 168
Approximations to the identity . . . . . . . . . . . . . . . . . . . . . . 170
Test functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Smooth partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . . 173
Convolutions of E-valued functions . . . . . . . . . . . . . . . . . . . . 177
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Linear differential operators . . . . . . . . . . . . . . . . . . . . . . . . 181
Weak derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
8 The substitution rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Pulling back the Lebesgue measure . . . . . . . . . . . . . . . . . . . . 191
The substitution rule: general case . . . . . . . . . . . . . . . . . . . . 195
Plane polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Polar coordinates in higher dimensions . . . . . . . . . . . . . . . . . . 198
Integration of rotationally symmetric functions . . . . . . . . . . . . . 202
The substitution rule for vector-valued functions . . . . . . . . . . . . . 203
9 The Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Definition and elementary properties . . . . . . . . . . . . . . . . . . . 206
The space of rapidly decreasing functions . . . . . . . . . . . . . . . . . 208
The convolution algebra S . . . . . . . . . . . . . . . . . . . . . . . . . 211
Calculations with the Fourier transform . . . . . . . . . . . . . . . . . 212
The Fourier integral theorem. . . . . . . . . . . . . . . . . . . . . . . . 215
Convolutions and the Fourier transform. . . . . . . . . . . . . . . . . . 218
Fourier multiplication operators . . . . . . . . . . . . . . . . . . . . . . 220
Plancherel’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
x Contents
Symmetric operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
The Heisenberg uncertainty relation . . . . . . . . . . . . . . . . . . . . 227
Chapter XI Manifolds and differential forms
1 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Definitions and elementary properties . . . . . . . . . . . . . . . . . . . 235
Submersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Submanifolds with boundary . . . . . . . . . . . . . . . . . . . . . . . . 246
Local charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Tangents and normals . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
The regular value theorem . . . . . . . . . . . . . . . . . . . . . . . . . 252
One-dimensional manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 256
Partitions of unity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
2 Multilinear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
Exterior products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
Pull backs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
The volume element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
The Riesz isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
The Hodge star operator . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Indefinite inner products . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
3 The local theory of differential forms . . . . . . . . . . . . . . . . . . . 285
Definitions and basis representations . . . . . . . . . . . . . . . . . . . 285
Pull backs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
The exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
The Poincar´elemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
4 Vector fields and differential forms . . . . . . . . . . . . . . . . . . . . 304
Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
Local basis representation . . . . . . . . . . . . . . . . . . . . . . . . . 306
Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
Local representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . 316
The exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Closed and exact forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
Orientability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
Tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
Contents xi
5 Riemannian metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
The volume element. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
The Hodge star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
The codifferential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
6 Vector analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
The Riesz isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
The divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
The Laplace–Beltramioperator . . . . . . . . . . . . . . . . . . . . . . 367
The curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
The Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
The Hodge–Laplace operator . . . . . . . . . . . . . . . . . . . . . . . . 379
The vector product and the curl . . . . . . . . . . . . . . . . . . . . . . 382
Chapter XII Integration on manifolds
1 Volume measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
The Lebesgue σ-algebra of M . . . . . . . . . . . . . . . . . . . . . . . 391
The definition of the volume measure . . . . . . . . . . . . . . . . . . . 392
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
Calculation of several volumes . . . . . . . . . . . . . . . . . . . . . . . 401
2 Integration of differential forms . . . . . . . . . . . . . . . . . . . . . . 407
Integrals of m-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
Restrictions to submanifolds . . . . . . . . . . . . . . . . . . . . . . . . 409
The transformation theorem . . . . . . . . . . . . . . . . . . . . . . . . 414
Fubini’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
Calculations of several integrals . . . . . . . . . . . . . . . . . . . . . . 418
Flows of vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
The transport theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
3 Stokes’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
Stokes’s theorem for smooth manifolds . . . . . . . . . . . . . . . . . . 430
Manifolds with singularities . . . . . . . . . . . . . . . . . . . . . . . . 432
Stokes’s theorem with singularities . . . . . . . . . . . . . . . . . . . . 436
Planar domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
Higher-dimensional problems . . . . . . . . . . . . . . . . . . . . . . . . 441
Homotopy invariance and applications . . . . . . . . . . . . . . . . . . 443
Gauss’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
Green’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
The classical Stokes’s theorem . . . . . . . . . . . . . . . . . . . . . . . 450
The star operator and the coderivative . . . . . . . . . . . . . . . . . . 451