A Course in Analysis

The book is an advanced textbook and a reference text in functional analysis in the wide sense. It provides advanced undergraduate and graduate students with a coherent introduction to the field, i.e. the basic principles, and leads them to more demanding topics such as the spectral theorem, Choquet theory, interpolation theory, analysis of operator semigroups, Hilbert–Schmidt operators and Hille–Tamarkin operators, topological vector spaces and distribution theory, fundamental solutions, or the Schwartz kernel theorem. All topics are treated in great detail and the text provided is suitable for self-studying the subject. This is enhanced by more than 270 problems solved in detail. At the same time the book is a reference text for any working mathematician needing results from functional analysis, operator theory or the theory of distributions. Embedded as Volume V in the Course of Analysis, readers will have a self-contained treatment of a key area in modern mathematics. A detailed list of references invites to further studies.Contents: Preface Introduction List of Symbols Functional Analysis: What is Functional Analysis about? Infinite Dimensional Vector Spaces Banach Spaces Linear Operators and Linear Functionals The Dual Space The Basic Principles of Functional Analysis Adjoint Operators and Fredholm Theory Hilbert Spaces and Operators in Hilbert Spaces Unbounded Operators in Hilbert Spaces Spectral Theory. Part I: Gelfand–Naimark Theory Spectral Theory. Part II: Self-adjoint Operators Topological Vector Spaces Convexity and Integral Representations Selected Topics Some Operator Theory: Some Integral Operators One-Parameter Semigroups of Operators Positivity Preserving Operators and Markovian Semigroups On Regular Sturm-Liouville Problems Sobolev Spaces. A First Encounter Operators Induced by the Dirichlet Problem Selected Topics Distributions: Some Function Spaces as Fréchet Spaces Distributions in the Sense of Schwartz Further Properties of Distributions Tempered Distributions and the Fourier Transform Tensor Products and the Kernel Theorem Calderon-Zygmund Operators Appendices: Completeness Nets, Convergence and Continuity On the Riesz Representation Theorem Solutions to Problems of Part 12 Solutions to Problems of Part 13 Solutions to Problems of Part 14 References Mathematicians Contributing to Analysis (Continued) Subject Index Readership: Advanced undergraduate students, graduate students, researchers in analysis. Functional Analysis;Hahn–Banach Theorems;Open Mapping Theorem;Closed Graph Theorem;Bounded and Unbounded Linear Operators;Spectral Theory;Convex Analysis;Krein–Milman Theorem;Choquet Theory;Bilinear Forms and Lax–milgram Theorem;Self-Adjoint Operators;Integral Operators;Hilbert–Schmidt Operators;Sturm–Liouville Problems;One–Parameter Semigroups of Operators;Markovian Semigroups;Interpolation Spaces;Riesz–Thorin and Marchinkiewicz Theorem;Unitary Representations Of Groups;Peter–Weyl Theorem;Topological Vector Spaces;Distributions in the Sense of L Schwartz;Fourier Transform of Tempered Distributions;Fundamental Solutions and Hypoellipticity;Kernel Theorem of L Schwartz0Key Features: In detail treatment of key topics in a core area of advanced analysis, suitable for self-studying A reference text in modern functional analysis and related topics with emphasis on solving operator equations More than 270 problems solved in detail Together with the other volumes of the Course, a coherent and comprehensive treatment of analysis