Analysis III

Foreword
Contents
Chapter IX Elements of measure theory
1 Measurable spaces
σ-algebras
The Borel σ-algebra
The second countability axiom
Generating the Borel σ-algebra with intervals
Bases of topological spaces
The product topology
Product Borel σ-algebras
Measurability of sections
2 Measures
Set functions
Measure spaces
Properties of measures
Null sets
3 Outer measures
The construction of outer measures
The Lebesgue outer measure
The Lebesgue–Stieltjes outer measure
Hausdorff outer measures
4 Measurable sets
Motivation
The σ-algebra of μ∗-measurable sets
Lebesgue measure and Hausdorff measure
Metric measures
5 The Lebesgue measure
The Lebesgue measure space
The Lebesgue measure is regular
A characterization of Lebesgue measurability
Images of Lebesgue measurable sets
The Lebesgue measure is translation invariant
A characterization of Lebesgue measure
The Lebesgue measure is invariant under rigid motions
The substitution rule for linear maps
Sets without Lebesgue measure
Chapter X Integration theory
1 Measurable functions
Simple functions and measurable functions
A measurability criterion
Measurable R-valued functions
The lattice of measurable R-valued functions
Pointwise limits of measurable functions
Radon measures
2 Integrable functions
The integral of a simple function
The L1-seminorm
The Bochner–Lebesgue integral
The completeness of L1
Elementary properties of integrals
Convergence in L1
3 Convergence theorems
Integration of nonnegative R-valued functions
The monotone convergence theorem
Fatou’s lemma
Integration of R-valued functions
Lebesgue’s dominated convergence theorem
Parametrized integrals
4 Lebesgue spaces
Essentially bounded functions
The Hölder and Minkowski inequalities
Lebesgue spaces are complete
Lp-spaces
Continuous functions with compact support
Embeddings
Continuous linear functionals on Lp
5 The n-dimensional Bochner–Lebesgue integral
Lebesgue measure spaces
The Lebesgue integral of absolutely integrable functions
A characterization of Riemann integrable functions
6 Fubini’s theorem
Maps defined almost everywhere
Cavalieri’s principle
Applications of Cavalieri’s principle
Tonelli’s theorem
Fubini’s theorem for scalar functions
Fubini’s theorem for vector-valued functions
Minkowski’s inequality for integrals
A characterization of Lp(Rm+n, E)
A trace theorem
7 The convolution
Defining the convolution
The translation group
Elementary properties of the convolution
Approximations to the identity
Test functions
Smooth partitions of unity
Convolutions of E-valued functions
Distributions
Linear differential operators
Weak derivatives
8 The substitution rule
Pulling back the Lebesgue measure
The substitution rule: general case
Plane polar coordinates
Polar coordinates in higher dimensions
Integration of rotationally symmetric functions
The substitution rule for vector-valued functions
9 The Fourier transform
Definition and elementary properties
The space of rapidly decreasing functions
The convolution algebra S
Calculations with the Fourier transform
The Fourier integral theorem
Convolutions and the Fourier transform
Fourier multiplication operators
Plancherel’s theorem
Symmetric operators
The Heisenberg uncertainty relation
Chapter XI Manifolds and differential forms
1 Submanifolds
Definitions and elementary properties
Submersions
Submanifolds with boundary
Local charts
Tangents and normals
The regular value theorem
One-dimensional manifolds
Partitions of unity
2 Multilinear algebra
Exterior products
Pull backs
The volume element
The Riesz isomorphism
The Hodge star operator
Indefinite inner products
Tensors
3 The local theory of differential forms
Definitions and basis representations
Pull backs
The exterior derivative
The Poincaré lemma
Tensors
4 Vector fields and differential forms
Vector fields
Local basis representation
Differential forms
Local representations
Coordinate transformations
The exterior derivative
Closed and exact forms
Contractions
Orientability
Tensor fields
5 Riemannian metrics
The volume element
Riemannian manifolds
The Hodge star
The codifferential
6 Vector analysis
The Riesz isomorphism
The gradient
The divergence
The Laplace–Beltrami operator
The curl
The Lie derivative
The Hodge–Laplace operator
The vector product and the curl
Chapter XII Integration on manifolds
1 Volume measure
The Lebesgue σ-algebra of M
The definition of the volume measure
Properties
Integrability
Calculation of several volumes
2 Integration of differential forms
Integrals of m-forms
Restrictions to submanifolds
The transformation theorem
Fubini’s theorem
Calculations of several integrals
Flows of vector fields
The transport theorem
3 Stokes’s theorem
Stokes’s theorem for smooth manifolds
Manifolds with singularities
Stokes’s theorem with singularities
Planar domains
Higher-dimensional problems
Homotopy invariance and applications
Gauss’s law
Green’s formula
The classical Stokes’s theorem
The star operator and the coderivative
References
Index