Functional Analysis Hardback
Part of the Graduate Studies in Mathematics series
Functional analysis is a central subject of mathematics with applications in many areas of geometry, analysis, and physics.
This book provides a comprehensive introduction to the field for graduate students and researchers.
It begins in Chapter 1 with an introduction to the necessary foundations, including the Arzela-Ascoli theorem, elementary Hilbert space theory, and the Baire Category Theorem.
Chapter 2 develops the three fundamental principles of functional analysis (uniform boundedness, open mapping theorem, Hahn-Banach theorem) and discusses reflexive spaces and the James space.
Chapter 3 introduces the weak and weak$^*$ topologies and includes the theorems of Banach-Alaoglu, Banach-Dieudonne, Eberlein-Smulyan, Krein-Milman, as well as an introduction to topological vector spaces and applications to ergodic theory.
Chapter 4 is devoted to Fredholm theory. It includes an introduction to the dual operator and to compact operators, and it establishes the closed image theorem.
Chapter 5 deals with the spectral theory of bounded linear operators.
It introduces complex Banach and Hilbert spaces, the continuous functional calculus for self-adjoint and normal operators, the Gelfand spectrum, spectral measures, cyclic vectors, and the spectral theorem.
Chapter 6 introduces unbounded operators and their duals.
It establishes the closed image theorem in this setting and extends the functional calculus and spectral measure to unbounded self-adjoint operators on Hilbert spaces.
Chapter 7 gives an introduction to strongly continuous semigroups and their infinitesimal generators.
It includes foundational results about the dual semigroup and analytic semigroups, an exposition of measurable functions with values in a Banach space, and a discussion of solutions to the inhomogeneous equation and their regularity properties.
The appendix establishes the equivalence of the Lemma of Zorn and the Axiom of Choice, and it contains a proof of Tychonoff's theorem.
With 10 to 20 elaborate exercises at the end of each chapter, this book can be used as a text for a one-or-two-semester course on functional analysis for beginning graduate students.
Prerequisites are first-year analysis and linear algebra, as well as some foundational material from the second-year courses on point set topology, complex analysis in one variable, and measure and integration.
- Format: Hardback
- Pages: 472 pages
- Publisher: American Mathematical Society
- Publication Date: 30/07/2018
- Category: Algebra
- ISBN: 9781470441906