Book:Peter D. Lax/Functional Analysis

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Peter D. Lax: Functional Analysis

Published $\text {2002}$, Wiley

ISBN 978-0471556046

Subject Matter

Functional Analysis



1. Linear Spaces
Axioms for linear spaces
Infinite dimensional examples
Subspace, linear span
Quotient space
Convex sets
Extreme subsets
2. Linear Maps
2.1 Algebra of linear maps
Axioms of linear maps
Sums and composites
Invertible linear maps
Nullspace and range
Invariant subspaces
2.2 Index of a linear map
Degenrate maps
Product formula for the index
Stability of the index
3. The Hahn-Banach Theorem
3.1 The extension theorem
Positive homogeneous, subadditive functionals
Extension of linear functionals
Gauge functions of convex sets
3.2 Geometric Hahn-Banach theorem
The hyperplane separation theorem
3.3 Extensions of Hahn-Banach theorem
The Agnew-Morse theorem
The Bohnenblust-Sobczyk-Soukhomlinov theorem
4. Applications of the Hahn-Banach Theorem
4.1 Extension of positive linear functionals
4.2 Banach limits
4.3 Finitely additive invariant set functions
Historical note
5. Normed Linear Spaces
5.1 Norms
Norms for quotient spaces
Complete normed linear spaces
The spaces $C$, $B$
$L^p$ spaces and Hölder's inequality
Sobolev spaces, embedding theorems
Separable spaces
5.2 Noncompactness of the unit ball
Uniform convexity
The Mazur-Ulam theorem on isometries
5.3 Isometries
6. Hilbert Space
6.1 Scalar product
Schwarz inequality
Parallelogram identity
Completeness, closure
6.2 Closest point in a closed convext subset
Orthogonal complement of a subspace
Orthogonal decomposition
6.3 Linear functionals
The Riesz-Frechet representation theorem
Lax-Milgram lemma
6.4 Linear span
Orthogonal projection
Orthonormal bases, Gram-Schmidt process
Isometries of Hilbert space
7. Applications of a Hilbert Space Results
7.1 Radon-Nikodym theorem
7.2 Dirichlet's problem
Use of the Riesz-Freceht theorem
Use of the Lax-Milgram theorem
Use of orthogonal decomposition
8. Duals of Normed Linear Spaces
8.1 Bounded linear functionals
Dual space
8.2 Extension of bounded linear functionals
Dual characterization of the norm
Dual characterization of distance from a subspace
Dual characterization of the closed linear span of a set
8.3 Reflexive spaces
Reflexivity of $L^p$, $1 < p < \infty$
Separable spaces
Separability of the dual
Dual of $\map C Q$, $Q$ compact
Reflexivity of subspaces
8.4 Support function of a set
Dual characterization of convex hull
Dual characterization of distance from closed, convex set
9. Applications of Duality
9.1 Completeness of weighted powers
9.2 The Müntz approxiation theorem
9.3 Runge's theorem
9.4 Dual variational problems in function theory
9.5 Existence of Green's function
10. Weak Convergence
10.1 Uniform boundedness of weakly converging sequences
Principle of uniform boundedness
Weakly sequentially closed convex sets
10.2 Weak sequential compactness
Compactness of unit ball in reflexive space
10.3 Weak* convergence
Helly's theorem
11. Applications of Weak Convergence
11.1 Approximation of the $\delta$ function by continuous functions
Toeplitz theorem on summability
11.2 Divergence of Fourier series
11.3 Approximate quadrature
11.4 Weak and strong analyticity of vector-valued functions
11.5 Existence of solutions of partial differential equations
Galerkin's method
11.6 The representation of analytics functions with positive real part
Herglotz-Riesz theorem
12. The Weak and Weak* Topologies
Comparison with sequential topology
Closed convex sets in the weak topology
Weak compactness
Alaoglu's theorem
13. Locally Convex Topologies and the Krein-Milman Theorem
13.1 Separation of points by linear functionals
13.2 The Krein-Milman theorem
13.3 The Stone-Weierstrass theorem
13.4 Choquet's theorem
14. Examples of Convex Sets and Their Extreme Points
14.1 Positive functionals
14.2 Convex functions
14.3 Completely monotone functions
14.4 Theorems of Caratheodory and Bochner
14.5 A theorem of Krein
14.6 Positive Harmonic functions
14.7 The Hamburger moment problem
14.8 G. Birkhoff's conjecture
14.9 De Finetti's theorem
14.10 Measure-preserving mappings
Historical note
15. Bounded Linear Maps
15.1 Boundedness and continuity
Norm of a bounded linear map
15.2 Strong and weak topologies
Strong and weak sequential convergence
15.3 Principle of uniform boundedness
15.4 Composition of bounded maps
15.5 The open mapping principle
Closed graph theorem
16. Examples of Bounded Linear Maps
16.1 Boundedness of integral operators
Integral operators of Hilbert-Schmidt type
Integral operators of Holmgren type
16.2 The convexity theorem of Marcel Riesz
16.3 Examples of bounded integral operators
The Fourier transform, Parseval's theorem and Hausdorff-Young inequality
The Hilbert transform
The Laplace transform
The Hilbert-Hankel transform
16.4 Solution operators for hyperbolic equations
16.5 Solution operator for the heat equation
16.6 Singular integral operators, pseudodifferential operators and Fourier integral operators
Fourier integral operators
17. Banach Algebras and their Elementary Spectral Theory
17.1 Normed algebras
Invertible elements
Resolvent set and spectrum
Spectral radius
17.2 Functional calculus
Spectral mapping theorem
18. Gelfand's Theory of Commutative Banach Algebras
Homomorphisms into $\C$
Maximal ideals
Mazur's lemma
The spectrum as the range of homomorphisms
The spectral mapping theorem revisited
The Gelfand representation
Gelfand topology
19. Applications of Gelfand's Theory of Commutative Banach Algebras
19.1 The algebra $\map C S$
19.2 Gelfand compactification
19.3 Absolutely convergent Fourier series
19.4 Analytic functions in the closed unit disk
Analytic functions in the polydisk
19.5 Analytic functions in the open unit disk
19.6 Wiener's Tauberian theorem
19.7 Commutative $\mathcal B^*$ algebras
Historical note
20. Examples of Operators and Their Spectra
20.1 Invertible maps
Boundary points of the spectrum
20.2 Shifts
20.3 Volterra integral operators
20.4 The Fourier transform
21. Compact Maps
21.1 Basic properties of compact maps
Compact maps form a two-sided ideal
Identity plus compact map has index zero
21.2 The spectral theory of compact maps
The transpose of a compact operator is compact
The Fredholm alternative
Historical note
22. Examples of Compact Operators
22.1 Compactness criteria
Arela-Ascoli compactness criterium
Rellich compactness criterium
22.2 Integral operators
Hilbert-Schmidt operators
22.3 The inverse of elliptic partial differential operators
22.4 Operators defined by parabolic equations
22.5 Almost orthogonal bases
23. Positive Compact Operators
23.1 The spectrum of compact positive operators
23.2 Stochastic integral operators
Invariant probability density
23.3 Inverse of a second order elliptic operator
24. Fredholm's Theory of Integral Equations
24.1 The Fredholm determinant and the Fredholm resolvent
The spectrum of Fredholm operators
A trace formula for Fredholm operators
24.2 Multiplicative property of the Fredholm determinant
24.3 The Gelfand-Levitan-Marchenko equation and Dyson's formula
25. Invariant Subspaces
25.1 Invariant subspaces of compact maps
The von Neumann-Aronszajn-Smith theorem
25.2 Nested invariant subspaces
Ringrose's theorem
Unicelullar operators: the Brodsky-Donoghue theorem
The Robinson-Bernstein and Lomonosov theorems
Enflo's example
26. Harmonic Analysis on a Halfline
26.1 The Phragmén-Lindelöf principle for harmonic functions
26.2 An abstract Phragmén-Lindelöf principle
Interior compactness
26.3 Asymptotic expansion
Solutions of elliptic differential equation in a half-cylinder
27. Index Theory
27.1 The Noether index
Stability of index
Product formula
Hörmander's stability theorem
Historical note
27.2 Toeplitz operators
Index-winding number
The inversion of Toeplitz operators
Discontinuous symbols
Matrix Toeplitz operators
27.3 Hankel operators
28. Compact Symmetric Operators in Hilbert Space
Variational principle for eigenvalues
Completness for eigenfunctions
The variational principles of Fisher and Courant
Functional calculus
Spectral theory of compact normal operators
Unitary operators
29. Examples of Compact Symmetric Operators
29.1 Convolution
29.2 The inverse of a differential operator
29.3 The inverse of partial differential operators
30. Trace Class and Trace Formula
30.1 Polar decomposition and singular values
30.2 Trace class, trace norm, trace
Matrix trace
30.3 The trace formula
Weyl's inequalities
Lidskii's theorem
30.4 The determinant
30.5 Examples and counterexamples of trace class operators
Mercer's theorem
The trace of integral operators
A Volterra integral operator
The trace of the powers of an operator
30.6 The Poisson summation formula
Convolution of $S^1$ and the convergence of Fourier series
The Selberg trace formula
30.7 How to express the index of an operator as a difference of traces
30.8 The Hilbert-Schmidt class
Relation of Hilbert-Schmidt class and trace class
30.9 Determinant and trace for operator in Banach spaces
31. Spectral Theory of Symmetric, Normal, and Unitary Operators
31.1 The spectrum of symmetric operators
Reality of spectrum
Upper and lower bounds for the spectrum
Spectral radius
31.2 Functional calculus for symmetric operators
The square root of a positive operator
Polar decomposition of bounded operators
31.3 Spectral resolution of symmetric operators
Projection-valued measures
31.4 Absolutely continuous, singular and point spectra
31.5 The spectral representation of symmetric operators
Spectral multiplicity
Unitary equivalence
31.6 Spectral resolution of normal operators
Functional calculus
Commutative $B^*$ algebras
31.7 Spectral resolution of unitary operators
Historical note
32. Spectral Theory of Self-Adjoint Operators
The Hellinger-Toeplitz theorem
Definition of self-adjointness
32.1 Spectral resolution
Sharpening of Herglotz's theorem
Cauchy transform of measures
The spectrum of self-adjoint operator
Representation of the resolvent as a Cauchy transform
Projection-valued measures
32.2 Spectral resolution using Cayley transform
32.3 A functional calculus for self-adjoint operators
33. Examples of Self-Adjoint Operators
33.1 The extension of unbounded symmetric operators
33.2 Examples of extension of symmetric operators; deficiency indices
The operator $i \paren{\dfrac d {d x} }$ on $\map {C_0^1} \R$, $\map {C_0^1} {\R_+}$ and $C_0^1 \paren {0, 1}$
Deficiency indices and von Neumann's theorem
Symmetric operators in a real Hilbert space
33.3 The Friedrichs extension
Semibounded symmetric operators
Symmetric ODE
Symmetric elliptic PDE
33.4 The Rellich perturbation theorem
Self-adjointness of Schrodinger operators with singular potentials
33.5 The moment problem
The Hamburger and Stieltjes moment problems
Uniqueness, or not, of the moment problem
Historical note
34. Semigroups of Operators
34.1 Strongly continuous one-parameter semigroups
Infinitesimal generator
Laplace transform
34.2 The generation of semigroups
The Hille-Yosida theorem
34.3 The approximation of semigroups
The Lax equivalence theorem
Trotter's product formula
Strang's product formula
34.4 Perturbation of semigroups
Lumer-Phillip's theorem
Trotter's perturbation theorem
34.5 The spectral theory if semigroups
Phillip's spectral mapping theorem
Adjoint semigroups
Semigroups of eventually compact operators
35. Groups of Unitary Operators
35.1 Stone's theorem
Generation of unitary groups
Positive definiteness and Bochner's theorem
35.2 Ergodic theory
von Neumann's mean ergodic theorem
35.3 The Koopman group
Volume-preserving flows
Metric transitivity
Time average
Space average
35.4 The wave equation
In full space-time
In the exterior of an obstacle
35.5 Translation representation
Sinai's theorem
Incoming subspaces
Solution of wave equation in odd number of space dimensions
Wave propagation outside an obstacle
35.6 The Heisenberg commutation relation
The uncertainty principle
Weyl's form of the commutation relation
von Neumman's theorem on pairs of operators that satisfy the commutation relation
Historical note
36. Examples of Strongly Continuous Semigroups
36.1 Semigroups defined by parabolic equations
36.2 Semigroups defined by elliptic equations
36.3 Exponential decay of semigroups
36.4 The Lax-Phillips semigroup
36.5 The wave equation in the exterior of an obstacle
37. Scattering Theory
37.1 Perturbation theory
37.2 The wave operators
37.3 Existence of the wave operators
37.4 The invariance of wave operators
37.5 Potential scattering
37.6 The scattering operator
Historical note
37.7 The Lax-Phillips scattering theory
37.8 The zeros of the scattering matrix
37.9 The automorphic wave equation
Faddeev and Pavlor's theory
The Riemann hypothesis
38. A Theorem of Beurling
38.1 The Hardy space
38.2 Beurling's theorem
Inner and outer factors
Factorization in the algebra of bounded analytic functions
38.3 The Titchmarsh convolution theorem
Historical note


A. A Riesz-Kakutani representation theorem
A.1 Positive linear functionals
A.2 Volume
A.3 $L$ as a space of functions
A.4 Measurable sets and measure
A.5 The Lebesgue measure and integral
B. Theory of distributions
B.1 Definitions and examples
B.2 Operations on distributions
B.3 Local properties of distributions
B.4 Applications to partial differential equations
B.5 The Fourier transform
B.6 Applications of the Fourier transform
B.7 Fourier series
C. Zorn's Lemma
Author Index
Subject Index