Functional Analysis, Spectral Theory, and Applications

Description

This textbook provides a careful treatment of functional analysis and some of its applications in analysis, number theory, and ergodic theory. In addition to discussing core material in functional analysis, the authors cover more recent and advanced topics, including Weyl’s law for eigenfunctions of the Laplace operator, amenability and property (T), the measurable functional calculus, spectral theory for unbounded operators, and an account of Tao’s approach to the prime number theorem using Banach algebras. The book further contains numerous examples and exercises, making it suitable for both lecture courses and self-study. Functional Analysis, Spectral Theory, and Applications is aimed at postgraduate and advanced undergraduate students with some background in analysis and algebra, but will also appeal to everyone with an interest in seeing how functional analysis can be applied to other parts of mathematics. Front Matter ....Pages i-xiv Motivation (Manfred Einsiedler, Thomas Ward)....Pages 1-14 Norms and Banach Spaces (Manfred Einsiedler, Thomas Ward)....Pages 15-70 Hilbert Spaces, Fourier Series, and Unitary Representations (Manfred Einsiedler, Thomas Ward)....Pages 71-120 Uniform Boundedness and the Open Mapping Theorem (Manfred Einsiedler, Thomas Ward)....Pages 121-133 Sobolev Spaces and Dirichlet’s Boundary Problem (Manfred Einsiedler, Thomas Ward)....Pages 135-166 Compact Self-Adjoint Operators and Laplace Eigenfunctions (Manfred Einsiedler, Thomas Ward)....Pages 167-208 Dual Spaces (Manfred Einsiedler, Thomas Ward)....Pages 209-252 Locally Convex Vector Spaces (Manfred Einsiedler, Thomas Ward)....Pages 253-312 Unitary Operators and Flows, Fourier Transform (Manfred Einsiedler, Thomas Ward)....Pages 313-352 Locally Compact Groups, Amenability, Property (T) (Manfred Einsiedler, Thomas Ward)....Pages 353-408 Banach Algebras and the Spectrum (Manfred Einsiedler, Thomas Ward)....Pages 409-431 Spectral Theory and Functional Calculus (Manfred Einsiedler, Thomas Ward)....Pages 433-485 Self-Adjoint and Symmetric Operators (Manfred Einsiedler, Thomas Ward)....Pages 487-502 The Prime Number Theorem (Manfred Einsiedler, Thomas Ward)....Pages 503-536 Back Matter ....Pages 537-614