Spring Semester 2009

Functional Analysis


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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
Office hours:  Mo, Fr 10:00 in Research I, 107
Grader:Mahmut Çalik

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
Additional reading TBA.


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

Last modified: 2009/04/16
This page: http://math.jacobs-university.de/oliver/teaching/jacobs/spring2009/math471/index.html
Marcel Oliver (m.oliver@jacobs-university.de)