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Functional Analysis Graduate Studies in Mathematics Graduate Studies in Mathematics 191 1st Edition by Theo Buhler, Dietmar A Salamon ISBN 147044190X 9781470441906

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Author(s): Theo Bühler, Dietmar A. Salamon
ISBN(s): 9781470441906, 147044190X
Edition: 1
File Details: PDF, 2.09 MB
Year: 2018
Language: english
SKU: EB-36177144 Category: Tags: ,

Functional Analysis Graduate Studies in Mathematics Graduate Studies in Mathematics 191 1st Edition by Theo Buhler, Dietmar A Salamon – Ebook PDF Instant Download/Delivery: 147044190X, 9781470441906
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ISBN 10: 147044190X 
ISBN 13: 9781470441906
Author: Theo Buhler, Dietmar A Salamon

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.

Functional Analysis Graduate Studies in Mathematics Graduate Studies in Mathematics 191 1st Table of contents:

  • Chapter 1 – Introduction to Functional Analysis

    • 1.1: Basic Concepts and Definitions

    • 1.2: Vector Spaces and Norms

    • 1.3: The Hahn-Banach Theorem

    • 1.4: The Open Mapping Theorem

    • 1.5: The Closed Graph Theorem

  • Chapter 2 – Banach Spaces

    • 2.1: Definition and Examples

    • 2.2: Continuous Linear Operators

    • 2.3: Dual Spaces

    • 2.4: Reflexive Banach Spaces

    • 2.5: Banach’s Fixed Point Theorem

  • Chapter 3 – Hilbert Spaces

    • 3.1: Definition and Examples

    • 3.2: Inner Products and Orthogonality

    • 3.3: The Riesz Representation Theorem

    • 3.4: Bounded Linear Operators on Hilbert Spaces

    • 3.5: Orthogonal Projections and the Projection Theorem

  • Chapter 4 – Operators on Banach and Hilbert Spaces

    • 4.1: Bounded Linear Operators

    • 4.2: Compact Operators

    • 4.3: The Spectrum of an Operator

    • 4.4: The Fredholm Alternative

    • 4.5: The Spectral Theorem

  • Chapter 5 – The Theory of Distributions

    • 5.1: Distributions and Test Functions

    • 5.2: Distributional Derivatives

    • 5.3: Convolution and Fourier Transforms

    • 5.4: Distributions in Sobolev Spaces

  • Chapter 6 – Operator Algebras

    • 6.1: C*-Algebras

    • 6.2: Unbounded Operators

    • 6.3: The Gelfand-Naimark Theorem

    • 6.4: Spectral Theory of Unbounded Operators

  • Chapter 7 – Applications of Functional Analysis

    • 7.1: Differential Equations and Functional Analysis

    • 7.2: Fixed Point Theorems and Applications

    • 7.3: Quantum Mechanics and Functional Analysis

    • 7.4: The Operator Norm and Applications in PDEs

  • Chapter 8 – Advanced Topics in Functional Analysis

    • 8.1: Nonlinear Functional Analysis

    • 8.2: Topological Vector Spaces

    • 8.3: The Baire Category Theorem

    • 8.4: Advanced Spectral Theory

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