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Unformatted text preview: Lectures in Functional Analysis Roman Vershynin Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI 48109, U.S.A. Email address : romanv@umich.edu Preface These notes are for a onesemester graduate course in Functional Analysis, which is based on measure theory. The notes correspond to the course Real Analysis II , which the author taught at University of Michigan in the Fall 2010. The course consists of about 40 lectures 50 minutes each. The student is assumed to be familiar with measure theory (both Lebesgue and abstract), have a good command of basic real analysis (epsilondelta) and abstract linear algebra (linear spaces and transformations). The course develops the theory of Banach and Hilbert spaces and bounded linear operators. Main principles of are covered in depth, which include Hahn Banach theorem, open mapping theorem, closed graph theorem, principle of uni form boundedness, and BanachAlaoglu theorem. Fourier series are developed for general orthogonal systems in Hilbert spaces. Compact operators and basics of Fredholm theory are covered. Spectral theory for bounded operators is studied in the second half of the course. This includes the spectral theory for compact selfadjoint operators, functional calculus and basic spectral theory of general (noncompact) operators, although the latter needs to be expanded a bit. Topics not covered include: KreinMilman theorem (although this can be done with one extra lecture), unbounded linear operators, and Fourier transform. Most applications to ODE and PDE are not covered, however the integral operators serve as a main example of operators in this course. The material has been compiled from several textbooks, including Eidelman, Milman and Tsolomitis Functional Analysis, Kirillov and Gvishiani Theorems and problemsin functional analysis, Reed and Simon Methods of modern mathe matical physics. I. Functional analysis, V. Kadets A course in functional analy sis (Russian), and P. Knyazev, Functional analysis. Minor borrowings are made from Yoshida Functional analysis, Rudin Functional analysis, and Conway A course in functional analysis. For some topics not covered, one may try R. Zimmer Essential results of functional analysis. Acknowledgement. The author is grateful to his students in the Math 602 course Real Analysis II , Winter 2010, who suggested numerous corrections for these notes. Special thanks are to Matthew Masarik for his numerous thoughtful remarks, corrections, and suggestions, which improved the presentation of this material. Contents Preface iv Chapter 1. Banach and Hilbert spaces 1 1.1. Linear spaces and linear operators 1 1.2. Normed spaces 7 1.3. Banach spaces 15 1.4. Inner product spaces 19 1.5. Hilbert spaces 25 1.6. Fourier series 28 Chapter 2. Bounded linear operators 39 2.1. Bounded linear functionals 39 2.2. Representation theorems for linear functionals 42 2.3. HahnBanach theorem 47...
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This note was uploaded on 12/08/2011 for the course MATH 623 taught by Professor Conlon during the Fall '08 term at University of Michigan.
 Fall '08
 CONLON
 Math

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