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Preface About thd Author Part I General Theory Topological Vector Spaces Introduction Separation properties Linear mappings Finite-dimensional spaces Metrization Boundedness and continuity Seminorms and local convexity Quotient spaces Examples Exercises 2 Completeness Baire category The Banach-Steinhaus theorem The open mapping theorem The closed graph theorem Bilinear mappings Exercises 3 Convexity The Hahn-Banach theorems Weak topologies Compact convex sets Vector-valued integration Holomorphic functions Exercises 4 Duality in Banach Spaces The normed dual of a normed space Adjoints Compact operators Exercises 5 Some Applications A continuity theorem Closed subspaces of fi-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 Part II Distributions and Fourier Transform 6 Test Functions and Distributions Introduction Test function spaces Calculus with distributions Localization Supports of distributions Distributions as derivatives Convolutions Exercises 7 Fourier Transforms Basic properties Tempered d]str]but]ons Paley-Wiener theorems Sobolev's lemma Exercises 8 Applications to Differential Equations Fundamental solutions Elliptic equations Exercises 9 Tauberian Theory Wiener's theorem The prime number theorem The renewal equation Exercises Part III Banach Algebras and Spectral Theory 10 Banach Algebras Introduction Complex homomorphisms Basic properties of spectra Symbolic calculus The group of invertible elements Lomonosov's invariant subspace theorem Exercises 11 Commutative Banach Algebras Ideals and homomorphisms Gelfand transforms Involutions Applications to noncommutative algebras Positive functionals Exercises 12 Bounded Operators on a Hilbert 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 13 Unbounded Operators Introduction Graphs and symmetric operators The Cayley transform Resolutions of the identity The spectral theorem Semigroups of operators Exercises Appendix A Compactness and Continuity Appendix B Notes and Comments Bibliography List of Special Symbols Index