Functional Analysis

Functional Analysis can mean different things, depending on who you ask. The core of the subject, however, is to study linear spaces with some topology which allows us to do analysis; ones like spaces of functions, spaces of operators acting on the space of functions, etc. Our interest in those spaces is twofold: those linear spaces with topology (i) often exhibit interesting properties that are worth investigating for their own sake, and (ii) have important application in other areas of mathematics (e.g., partial differential equations) as well as theoretical physics; in particular, quantum mechanics. (i) arises because linear vectors spaces that are of interest to analysts are infinite-dimensional in nature, and this requires careful investigation of geometry. (More on this in Chapter 2 and 4.) (ii) was what initially motivated the development of the field; Functional Analysis has its historical roots in linear algebra and the mathematical formulation of quantum mechanics in the early 20 century. (See w:Mathematical formulation of quantum mechanics) The book aims to cover these two interests simultaneously.

The book consists of two parts. The first part covers the basics of Banach spaces theory with the emphasis on its applications. The second part covers topological vector spaces, especially locally convex ones, generalization of Banach spaces. In both parts, we give principal results e.g., the closed graph theorem, resulting in some repetition. One reason for doing this organization is that one often only needs a Banach-version of such results. Another reason is that this approach seems more pedagogically sound; the statement of the results in their full generality may obscure its simplicity. Exercises are meant to be an unintegrated part of the book. They can be skipped altogether, and the book should be fully read and understood. Some alternative proofs and additional results are relegated as exercises when their inclusion may disrupt the flow of the exposition.

Knowledge of measure theory will not be needed except for Chapter 6, where we formulate the spectrum theorem in the language of measure theory. As for topology, knowledge of metric spaces suffices for Chapter 1 and Chapter 2. The solid background in general topology is required for the ensuing chapters.

Contents

Part 1:

Chapter 1. Preliminaries

Zorn's lemma, Topology, Hamel basis, Hahn-Banach theorem

Chapter 2. Banach spaces

Open mapping theorem, closed graph theorem, compact operators

Chapter 3. Hilbert spaces

Unbounded operators, adjoint operators, orthonormal basis, Parseval theorem

Chapter 4. Geometry of Banach spaces

Reflexive spaces, Krein-Milman theorem, Bishop's theorem, Separable Banach spaces, Schauder basis, James' theorem, uniformly convex spaces, monotonic operators, strictly singular operators

Part 2:

Chapter 5. Topological vector spaces

Locally convex spaces, metrization theorem

Chapter 6. C*-algebras

Gelfand transformation, Spectrum of a commutative Banach algebra, Functional calculus, GNS construction

Chapter 7. Integration theory

von Neumann double centralizer theorem

Part 3:

Chapter 8. Special topics

• Fredholm theory - Index of a bounded and unbounded operator, compact perturbation

  Representations of compact groups - Unitary representation, Peter–Weyl theorem

  Sobolev spaces

Chapter 9. Harmonic Analysis - Decomposition of Banach Representations of Locally compact groups, Plancherel theorem.