泛函分析(英文版原书第2版典藏版)/华章数学原版精品系列

本书是国际著名教材，在材料的取舍和处理手法 上很有特色，对某些公理进行了准确描述，并精彩地 讨论了一些深入的专题，还介绍了在其他数学分支 （如微分方程）中有价值的应用。用作者自己的话来 讲，他并不期望写一部百科全书，而是为进一步的探 索打开通道。 本书叙述清楚，论证严谨，不少地方的注释相当 精辟并具有启发性，可作为高等院校数学专业高年级 本科生和研究生的教材和参考书。

Preface About the Author Part I General Theory 1 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 Lp-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 Transforms 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 distributions 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