Preface |
General Theory / Part 1: |
Topological Vector Space Introduction Separation properties Linear Mappings Finite-dimensional spaces Metrization Boundedness and continuity Seminorms and local convexity Quotient spaces Examples Exercises / 1: |
Completeness Baire category The Banach-Steinhaus theorem The open mapping theorem The closed graph theorem Bilinear mappings Exercises / 2: |
Convexity The Hahn-Banach theorems Weak topologies Compact convex sets Vector-valued integration Holomorphic functions Exercises / 3: |
Duality in Banach Spaces The normed dual of a normed space Adjoints Compact operators Exercises / 4: |
Some Applications A continuity theorem Closed subspaces ofL p -spaces The range of a vector-valued measure A generalized Stone-Weierstrass theorem Two interpolation theorems Kakutani's fixed point theorem Haar measure on compact groups Uncomplemented subspaces Sums of Poisson kernels Two more fixed point theorems Exercises / 5: |
Distributions and Fourier Transforms / Part 2: |
Test Functions and Distributions Introduction Test function spaces Calculus with distributions Localization Supports of distributions Distributions as derivatives Convolutions Exercises / 6: |
Fourier Transforms Basic properties Tempered distributions Paley-Wiener theorems Sobolev's lemma Exercises / 7: |
Applications to Differential Equations Fundamental solutions Elliptic equations Exercises / 8: |
Tauberian Theory Wiener's theorem The prime number theorem The renewal equation Exercises / 9: |
Banach Algebras and Spectral Theory / Part 3: |
Banach Algebras Introduction Complex homomorphisms Basic properties of spectra Symbolic calculus The group of invertible elements Lomonosov's invariant subspace theorem Exercises / 10: |
Commutative Banach Algebras Ideals and homomorphisms Gelfand transforms Involutions Applications to noncommutative algebras Positive functionals Exercises / 11: |
Bounded Operators on a Hillbert Space Basic facts Bounded operators A commutativity theorem Resolutions of the identity The spectral theorem Eigenvalues of normal operators Positive operators and square roots The group of invertible operators A characterization of B*-algebras An ergodic theorem Exercises / 12: |
Unbounded Operators Introduction Graphs and symmetric operators The Cayley transform Resolutions of the identity The spectral theorem Semigroups of operators Exercises / 13: |
Compactness and Continuity / Appendix A: |
Notes and Comments / Appendix B: |
Bibliography List of Special Symbols |
Index |
Preface |
General Theory / Part 1: |
Topological Vector Space Introduction Separation properties Linear Mappings Finite-dimensional spaces Metrization Boundedness and continuity Seminorms and local convexity Quotient spaces Examples Exercises / 1: |