Functional Analysis

Syllabus

Summary:
This course covers the basic theorems of functional analysis and provides an introduction to linear operators on Banach spaces. Starting with the abstract notion of topological spaces, we re-develop some of the concepts which were already introduced in Real Analysis in an abstract Banach space setting. The course further contains an introduction to the theory of linear operators, including spectral theory for compact operators, with applications to integral operators and boundary value problems.

Contact Information:
 Instructor: Marcel Oliver Email: m.oliver@jacobs-university.de Phone: 200-3212 Office hours: Mo, Fr 10:00 in Research I, 107 Grader: Mahmut Çalik Email: m.calik@jacobs-university.de

Time and Place:
 Lectures: We 11:15, Fr 9:45 in East Hall 3

Recommended Textbook:
G.B. Folland: Real Analysis, second edition, Wiley, 1999

• The final grade will be computed as a grade point average with the following weights:

 Homework: 30% Midterm Exam: 30% Final Exam: 40%

• Each of the individual scores will be converted to Jacobs grade points before the overall weighted grade point average is computed.

Class Schedule (subject to change!)

 02/02/2009: Point-set topology - topological spaces 04/04/2009: Point-set topology - continuity, product topology 09/02/2009: Spaces of bounded and of continuous functions, Urysohn's lemma; compact spaces 11/02/2009: Nets, Tychonoff's theorem 16/02/2009: Arzela-Ascoli theorem; Banach spaces, the Hahn-Banach theorem 18/02/2009: Geometric forms of the Hahn-Banach theorem; separation of convex subsets 23/02/2009: Baire category theorem, open mapping theorem 25/02/2009: Closed graph theorem; uniform boundedness principle 02/03/2009: Topological vector spaces 04/03/2009: No class 09/03/2009: Weak topologies, the weak and the weak-* topology; Banach-Alaoglu theorem 11/03/2009: Topologically complementary subspaces I 16/03/2009: Topologically complementary subspaces II 18/03/2009: Unbounded operators, adjoints, invertibility 20/03/2009: Midterm review 23/03/2009: Midterm Exam 25/03/2009: Extensions, closeable operators; examples of differential operators 30/03/2009: No class 01/04/2009: No class 15/04/2009: Examples of differential operators and their adjoints 17/04/2009: Introductory remarks on spectral theory; definitions; 9:00 in Research I, 107 20/04/2009: Fredholm alternative 22/04/2009: Spectrum of a compact operator 24/04/2009: Lax-Milgram Theorem; residual spectrum, self-adjoint operators; spectral theorem for compact, self-adjoint operators; 9:00 in Research I, 107 27/04/2009: Examples: Hilbert-Schmidt operators, inverses of unbounded operators on a Gelfand-triple are compact 29/04/2009: Interlude: Introduction to Sobolov spaces 04/05/2009: Invertibility of unbounded operators revisited; the spectrum of an unbounded operator 06/05/2009: Spectral mapping theorem 08/05/2009: Review for final exam 9:00 in Research I, 107 11/05/2009: No class 13/05/2009: No class 20/05/2009: Final Exam, 8:00-10:00, East Hall 2