COURSE: MATH 5740-001

TIME: 11:20-12:50 MTWRF, PLACE: Room 314 Gilbreath Hall

INSTRUCTOR: Dr. Robert Gardner, OFFICE: Room 308F of Gilbreath Hall

OFFICE HOURS: TBA PHONE: 439-6979 (Math Office 439-4349)

E-MAIL: gardnerr@etsu.edu
WEBPAGE: http://faculty.etsu.edu/gardnerr/gardner.htm

TEXT: A First Course in Functional Analysis, by S. David Promislow, John Wiley & Sons Publications (2008). Also, Chapter 5 of Real Analysis with an Introduction to Wavelets and Applications, by D. Hong, J. Wang, and R. Gardner, Academic Press/Elsevier Press (2005) will be used as a supplement.

ABOUT THE COURSE: At this stage, this class is technically an "experimental class" which is not yet on the books, but after it is taught a couple of times it will become an official class which appears in the catalog. The catalog description will be: "An introduction to the standard topics of functional analysis are given. Properties of normed linear spaces, Banach spaces, and Hilbert spaces are studied. The Hahn-Banach Theorem is addressed. Spectral theory and compact operators are introduced. A knowledge of measure theory is not assumed and any needed measure theoretic results are presented in the course." As the last sentence suggests, a functional analysis class normally has a prerequisite of a graduate level real analysis sequence (ETSU's Real Analysis 1 and 2, MATH 5210-5220). So a full-blown functional analysis sequence appropriately belongs in a Ph.D. program. This is why our class is titled "Introduction to Functional Analysis."

ONLINE NOTES: Online notes in PDF form are available for each section we cover. The notes include definitions, some motivational comments, and statements of lemmas, theorems, and corollaries, but mostly omit the detailed proofs. The notes can be found at:

GRADING: Homework will be assigned on a regular basis and your grade on the homework will determine your grade for the course. Grades will be assigned based on a 10 point scale with "plus" and "minus" grades being assigned as appropriate. Based on the assignment of grad points by ETSU, the plus and minus grades should be given on a 3 point subscale. For example, a B+ corresponds to an average of 87, 88, or 89; an A- corresponds to an average of 90, 91, or 92; an A corresponds to an average of 93 to 100 (ETSU does not grant A+ grades, nor D- grades), etc.

A NOTE ABOUT HOMEWORK: While I suspect that you may work with each other on the homework problems (in fact, I encourage you to), I expect that the work you turn in is your own and that you understand it. If I get homework from two (or more) of you that is virtually identical, then neither of you will get any credit. Some of the homework problems are fairly standard for this class, and you may find proofs online. However, the online proofs may not be done with the notation, definitions, and specific methods which we are developing and, therefore, are not acceptable for this class. You are expected to give all details and document all claims on the homework!!!

TENTATIVE OUTLINE:
We will attempt to cover topics from this list:
Chapter 1: Linear Spaces and Operators.
Preliminary definitions and examples, linear operators, null space, spaces of linear operators, finite and infinite dimensional spaces.
Chapter 2: Normed Linear Spaces: The Basics.
Metric spaces, norms, open and closed sets, continuity and uniform continuity, compact sets, bounded linear operators, completeness, Banach spaces, Extension Theorem, completion of a normed linear space, comparison of norms, quotient spaces topological criterion for finite dimensionality, L^p space, l ^p space, survey of Lebesgue measure and integration, Schauder bases and Hamel bases, direct products and sums, contraction mappings.
Chapter 3: Major Banach Space Theorems.
Baire's Theorem, Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Principle.
Vector Spaces, Hilbert Spaces, and the L² Space (Hong, Wang and Gardner; partial).
Groups, fields, and vector spaces, inner product spaces, L², projections and Hilbert space isomorphisms, the Fundamental Theorem of Infinite Dimensional Vector Spaces.
Chapter 4: Hilbert Spaces.
Inner products and semi-inner products, Cauchy-Schwartz Inequality, Triangle Inequality, Parallelogram Law, Polarization Identity, nearest points and convex sets, orthogonality, Projection Theorem, orthonormal sets, Gram-Schmidt Process, linear functionals on Hilbert spaces, linear operators on Hilbert spaces, sesquilinear forms, Hilbert space adjoints, normal operators, self-adjoint operators, positive operators, projections, unitary operators, ordering self-adjoint operators.
Chapter 5: Hahn-Banach Theorem.
The Real Hahn-Banach Extension Theorem, Complex Hahn-Banach Extension Theorem, Normed Linear Space Version, linear manifolds, half spaces and separations, internal points, bounding points, Minkowski functionals, Geometric Hahn-Banach Extension Theorem, Hahn-Banach Separation Theorem, supporting hyperplanes.
Chapter 6: Duality. (Time Permitting)
Duals of the l^p space, the Radon-Nikodym Theorem, signed measures, absolutely continuous measures, dual of L^p, dual of C[a,b], Riesz Representation Theorem, adjoints, double duals, reflexivity, General Uniform Boundedness Principle, weak convergence, weak* convergence, sequential convergence.
Chapter 8: The Spectrum.(Time Permitting)
Spectrum, Banach algebras, Spectral Mapping Theorem, spectrum of a normal operator, eigenvalues and eigenspaces, C* algebras.
Chapter 9: Compact Operators.(Probably not)
Relatively compact sets, totally bounded set, equicontinuity, Arzela-Ascoli Theorem, building compact operators, spectrum of compact operators, compact self-adjoint operators, Spectral Theorem for compact Self-Adjoint Operators, invariant subspaces.

TENTATIVE SCHEDULE: This is ambitious!
Week 1: Chapter 1, Chapter 2.
Week 2: Chapter 2, Chapter 3.
Week 3: Chapter 3, Chapter 4, plus supplements.
Week 4: Chapter 4, plus supplements.
Week 5: Chapter 5, Chapter 6, Chapter 8.

IMPORTANT DATES: Official dates are online at:

1. Michael Reed and Barry Simon, Methods of Modern Mathematical Physics: I. Functional Analysis (Revised and Enlarged Edition), Academic Press (1980). This is a standard graduate level text on functional analysis (in fact, it was used at Auburn University in the late 1980s). The Sherrod Library has a copy of the 1972 version - it's call number is QC20.R37 1972. It is referred to in the class notes as simply "Reed and Simon's Functional Analysis."
2. Lokenath Debnath and Piotr Mikusinski, Introduction to Hilbert Spaces with Applications 3rd Edition, Elsevier Academic Press (2005). This book includes much of the material we cover on Hilbert spaces and operators on Hilbert spaces. It also includes applications to integral and differential equations, partial differential equations, quantum mechanics, and wavelets.
3. The notes for the "Meaning of Mathematics" lecture are also online at: Meaning of Mathematics. These notes are of a philosophical nature and contain basic knowledge that every math major should know.

OTHER RESOURCES. The following were mentioned in class:

1. To access the Mathematical Reviews: Go to the Sherrod Library online catalog. Click the "Title" tab and enter "Mathematical Reviews." Select "MathSciNet [Electronic Resource]" and follow the links. You will be asked to enter your user ID and password (the same you use for your ETSU e-mail). You are then redirected to MathSciNet and can freely use it and even download PDF versions of some of the papers you find! Of course, this protocol will work for any electronic journal available through the Sherrod Library.

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Last updated: July 8, 2015.