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
 * Lp spaces and Holder'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 Muntz 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 convertgent 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 Phragmen-Lindelof principle for harmonic functions
 * 26.2 An abstract Phragmen-Lindelof 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
 * Hormander'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


 * 32. Spectral Theory of Self-Adjoint Operators


 * 33. Examples of Self-Adjoint Operators


 * 34. Semigroups of Operators


 * 35. Groups of Unitary Operators


 * 36. Examples of Strongly Continuous Semigroups


 * 37. Scattering Theory


 * 38. A Theorem of Beurling

Texts


 * A. A Riesz-Kakutani representation theorem


 * B. Theory of distributions


 * C. Zorn's Lemma


 * Author Index


 * Subject Index