Book:Peter D. Lax/Functional Analysis

Subject Matter

 * Functional Analysis

Contents
Foreword


 * 1. Linear Spaces


 * Axioms for linear spaces
 * Infinite dimensional examples
 * Subspace, linear span
 * Quotient space
 * Isomorphism
 * 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
 * Pseudoinverse
 * Index
 * 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
 * $l^2,L^2$


 * 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
 * Transpose


 * 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
 * Resolvent
 * Spectral radius


 * 17.2 Functional calculus


 * Spectral mapping theorem
 * Projections


 * 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


 * Pseudoinverse
 * 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
 * Domain


 * 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
 * Resolvent
 * 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

Texts


 * 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