Analysis III

Analysis III......Page 3
Foreword......Page 5
Foreword to the English translation......Page 6
Chapter IX Elements of measure theory......Page 7
Chapter X Integration theory......Page 8
Chapter XI Manifolds and differential forms......Page 10
Chapter XII Integration on manifolds......Page 11
References......Page 12
Chapter IX: Elements of measure theory......Page 13
σ-algebras......Page 15
The Borel σ-algebra......Page 17
The second countability axiom......Page 18
Generating the Borel σ-algebra with intervals......Page 20
Bases of topological spaces......Page 21
The product topology......Page 22
Product Borel σ-algebras......Page 24
Measurability of sections......Page 25
Set functions......Page 29
Properties of measures......Page 30
Null sets......Page 32
The construction of outer measures......Page 36
The Lebesgue outer measure......Page 37
The Lebesgue–Stieltjes outer measure......Page 40
Hausdorff outer measures......Page 41
Motivation......Page 44
The σ-algebra of µ*-measurable sets......Page 45
Lebesgue measure and Hausdorff measure......Page 47
Metric measures......Page 48
The Lebesgue measure space......Page 52
The Lebesgue measure is regular......Page 53
Images of Lebesgue measurable sets......Page 56
The Lebesgue measure is translation invariant......Page 59
A characterization of Lebesgue measure......Page 60
The Lebesgue measure is invariant under rigid motions......Page 62
The substitution rule for linear maps......Page 63
Sets without Lebesgue measure......Page 65
Chapter X: Integration theory......Page 70
Simple functions and measurable functions......Page 73
A measurability criterion......Page 75
Measurable R-valued functions......Page 78
The lattice of measurable R-valued functions......Page 79
Pointwise limits of measurable functions......Page 84
Radon measures......Page 85
The integral of a simple function......Page 91
The L1-seminorm......Page 93
The Bochner–Lebesgue integral......Page 95
The completeness of L1......Page 98
Elementary properties of integrals......Page 99
Convergence in L1......Page 102
Integration of nonnegative R-valued functions......Page 108
The monotone convergence theorem......Page 111
Fatou’s lemma......Page 112
Integration of R-valued functions......Page 114
Lebesgue’s dominated convergence theorem......Page 115
Parametrized integrals......Page 118
Essentially bounded functions......Page 121
The Hölder and Minkowski inequalities......Page 122
Lebesgue spaces are complete......Page 125
Lp-spaces......Page 127
Continuous functions with compact support......Page 129
Embeddings......Page 130
Continuous linear functionals on Lp......Page 132
Lebesgue measure spaces......Page 139
The Lebesgue integral of absolutely integrable functions......Page 140
A characterization of Riemann integrable functions......Page 143
Maps defined almost everywhere......Page 148
Cavalieri’s principle......Page 149
Applications of Cavalieri’s principle......Page 152
Tonelli’s theorem......Page 155
Fubini’s theorem for scalar functions......Page 156
Fubini’s theorem for vector-valued functions......Page 159
Minkowski’s inequality for integrals......Page 163
A characterization of Lp(R m+n,E)......Page 168
A trace theorem......Page 169
Defining the convolution......Page 173
The translation group......Page 176
Elementary properties of the convolution......Page 179
Approximations to the identity......Page 181
Test functions......Page 183
Smooth partitions of unity......Page 184
Distributions......Page 188
Linear differential operators......Page 192
Weak derivatives......Page 195
Pulling back the Lebesgue measure......Page 202
The substitution rule: general case......Page 206
Plane polar coordinates......Page 208
Polar coordinates in higher dimensions......Page 209
Integration of rotationally symmetric functions......Page 213
The substitution rule for vector-valued functions......Page 214
Definition and elementary properties......Page 217
The space of rapidly decreasing functions......Page 219
The convolution algebra S......Page 222
Calculations with the Fourier transform......Page 223
The Fourier integral theorem......Page 226
Convolutions and the Fourier transform......Page 229
Fourier multiplication operators......Page 231
Plancherel’s theorem......Page 234
Symmetric operators......Page 236
The Heisenberg uncertainty relation......Page 238
Chapter XI: Manifolds and differential forms......Page 243
Definitions and elementary properties......Page 245
Submersions......Page 251
Submanifolds with boundary......Page 256
Local charts......Page 260
Tangents and normals......Page 261
The regular value theorem......Page 262
Partitions of unity......Page 266
Exterior products......Page 270
Pull backs......Page 277
The volume element......Page 278
The Riesz isomorphism......Page 281
The Hodge star operator......Page 283
Indefinite inner products......Page 287
Tensors......Page 291
Definitions and basis representations......Page 295
Pull backs......Page 299
The exterior derivative......Page 302
The Poincaré lemma......Page 305
Tensors......Page 309
Vector fields......Page 314
Local basis representation......Page 316
Differential forms......Page 318
Local representations......Page 321
Coordinate transformations......Page 326
The exterior derivative......Page 329
Closed and exact forms......Page 331
Contractions......Page 332
Orientability......Page 334
Tensor fields......Page 340
The volume element......Page 343
Riemannian manifolds......Page 347
The Hodge star......Page 358
The codifferential......Page 360
The Riesz isomorphism......Page 368
The gradient......Page 371
The divergence......Page 373
The Laplace–Beltrami operator......Page 377
The curl......Page 382
The Lie derivative......Page 385
The Hodge–Laplace operator......Page 389
The vector product and the curl......Page 392
Chapter XII: Integration on manifolds......Page 399
The Lebesgue σ-algebra of M......Page 401
The definition of the volume measure......Page 402
Properties......Page 407
Integrability......Page 408
Calculation of several volumes......Page 411
Integrals of m-forms......Page 417
Restrictions to submanifolds......Page 419
The transformation theorem......Page 424
Fubini’s theorem......Page 425
Calculations of several integrals......Page 428
Flows of vector fields......Page 431
The transport theorem......Page 435
Stokes’s theorem for smooth manifolds......Page 440
Manifolds with singularities......Page 442
Stokes’s theorem with singularities......Page 446
Planar domains......Page 449
Higher-dimensional problems......Page 452
Homotopy invariance and applications......Page 453
Gauss’s law......Page 456
Green’s formula......Page 458
The classical Stokes’s theorem......Page 460
The star operator and the coderivative......Page 462
References......Page 466
Index......Page 468
Sachverzeichnis......Page 0