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


 * 8. Duals of Normed Linear Spaces


 * 9. Applications of Duality


 * 10. Weak Convergence


 * 11. Applications of Weak Convergence


 * 12. The Weak and Weak* Topologies


 * 13. Locally Convex Topologies and the Krein-Milman Theorem


 * 14. Examples of Convex Sets and Their Extreme Points


 * 15. Bounded Linear Maps


 * 16. Examples of Bounded Linear Maps


 * 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