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 50% October, 2009 Zorn's lemma, Topology, Hamel basis, Hahn-Banach theorem Chapter 2. Banach spaces 50% Sep 1, 2007 Open mapping theorem, closed graph theorem, compact operators Chapter 3. Hilbert spaces 75% June 4, 2008 Unbounded operators, adjoint operators, orthonormal basis, Parseval theorem Chapter 4. Geometry of Banach spaces 00% May 27, 2008 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 25% May 28, 2008 Locally convex spaces, metrization theorem Chapter 6. C*-algebras 00% October 30, 2008 Gelfand transformation, Spectrum of a commutative Banach algebra, Functional calculus, GNS construction Chapter 7. Integration theory 00% August 16, 2009 von Neumann double centralizer theorem Part 3: Chapter 8. Special topics 00% June 6, 2008 Fredholm theory 25% May 10, 2009 - Index of a bounded and unbounded operator, compact perturbation Representations of compact groups 00% June 30, 2009 - Unitary representation, Peter–Weyl theorem Sobolev spaces 00% October, 2009 Chapter 9. Harmonic Analysis 00% September 23, 2018 - Decomposition of Banach Representations of Locally compact groups, Plancherel theorem. Mathematical analysis University level mathematics books F 25%