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


 * 18. Gelfand's Theory of Commutative Banach Algebras


 * 19. Applications of Gelfand's Theory of Commutative Banach Algebras


 * 20. Examples of Operators and Their Spectra


 * 21. Compact Maps


 * 22. Examples of Compact Operators


 * 23. Positive Compact Operators


 * 24. Fredholm's Theory of Integral Equations


 * 25. Invariant Subspaces


 * 26. Harmonic Analysis on a Halfline


 * 27. Index Theory


 * 28. Compact Symmetric Operators in Hilbert Space


 * 29. Examples of Compact Symmetric Operators


 * 30. Trace Class and Trace Formula


 * 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