Einsiedler M., Ward T. Functional Analysis, Spectral Theory, and Applications

New York: Springer, 2017. — 626 p.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.Motivation
Norms and Banach Spaces
Hilbert Spaces, Fourier Series, and Unitary Representations
Uniform Boundedness and the Open Mapping Theorem
Sobolev Spaces and Dirichlet’s Boundary Problem
Compact Self-Adjoint Operators and Laplace Eigenfunctions
Dual Spaces
Locally Convex Vector Spaces
Unitary Operators and Flows, Fourier Transform
Locally Compact Groups, Amenability, Property (T)
Banach Algebras and the Spectrum
Spectral Theory and Functional Calculus
Self-Adjoint and Symmetric Operators
The Prime Number Theorem