Math 8801-8802: Fall 2007/Spring 2008

Math 8801-8802: Functional Analysis

Homework 1 , Homework 2 , Midterm 1
Homework 3 , Homework 4 , Midterm 2

Homework 5 , Homework 6 , Homework 7

Instructor. Marta Lewicka (office hours in VinH 328, Wednesdays from 2:30pm)
Recommended texts. The course will be self-contained, however the following books are recommended and should be frequently consulted:
- Brezis: Analyse fonctionelle - Theorie et applications
- Megginson: An introduction to Banach space theory
- Rudin: Functional analysis
- Kirillov and Gvishiani: Theorems and problems in functional analysis
- Halmos: A Hilbert space problem book
- Lax: Functional analysis
- Yoshida: Functional Analysis
- Albiac and Kalton: Topics in Banach space theory
- Lloyd: Degree theory
Prerequisites. The course is intended for graduate students; undergraduates require Math 5615-5616 and the permission of the instructor.
Grades for 8801. Grades will be based on bi-weekly homework (50%) and two midterms (50%).
Grades for 8802. Grades will be based on homework and a presentation. List of topics.
Homework. Homework will be assigned on Fridays, and due on Friday after two weeks at the beginning of class. Late homework will not be accepted. The solution of each exercise will be evaluated in the scale 0-10 points. You should write up solutions independently.
Topics.
1. Review of basic structures and examples,
2. Hahn-Banch theorem: analytic form, real and complex versions, the distance formula, spanning criterion and applications, extension to families of linear maps, finitely additive invariant set functions, geometric form, separation theorems, Banach limits, Muntz approximation theorem,
3. Open mapping theorem: Baire's lemma, Banach-Steinhaus theorem, open mapping and closed graph theorems,
4. Complementarity: complemented spaces, projections, one-sided inverses, example of a non-complemented space, orthogonality relations,
5. The adjoint operator: unbounded operators, the adjoint, relations between surjectivity/injectivity of operator and its adjoint, method of a-priori bounds, applications,
6. Weak topologies: abstract setting, weak and weak* topologies and convergences, weak, weak* and strong convexity, continuity, compactness, Banach-Alaoglu's theorem, Kakutani's theorem,Mazur's theorem, reflexivity, separability, metrizability of weak topologies, uniformly convex spaces,
7. Compact operators: Schauder fixed point theorem, finite rank operators, Schauder's theorem, Hilbert-Schmidt operators, Schur's operators, Fredholm alternative, spectrum of a compact operator, spectral theory of compact self-adjoint operators, Courant-Fischer formulas, spectrum of a compact positive oparator,
8. Fredholm index theory: pseudoinverses, stability, homotopy invariance,
9. Lax-Milgram and Stampacchia theorems, variational inequalities and minimization problem, Hartman-Stampacchia theorem,
10. Hille-Yoshida theorem for maximal monotone operators in Hilbert spaces: Yoshida regularisation, evolutionary problem for unbounded operators, Hille-Yoshida theorem , higher regularity, the self-adjoint case, smoothing, accretive operators and Hille-Yoshida thm in Banach spaces, application to heat and wave equations,
11. Continuous semigroups of operators in Banach spaces: infinitesimal generator, type of semigroup, Bochner integral, the Hille-Yoshida theorem, Phillips' theorem, contractive semigroups, Lions' approach to defining continuous semigroups,
12. The Brouwer topological degree: the Amann-Weiss axioms, construction of the degree via sgn of determinants, Sard's lemma, proof of homotopy invariance and other axioms, alternative formulas for the degree, the winding number,
13. Applications of Brouwer's degree: Brouwer fixed point theorem, Perron-Frobenius theorem, Poincare'-Brouwer theorem and combing a hairy ball, the odd mapping theorem, Borsuk's antipodal theorem, Borsuk-Lusternik-Schnirelmann theorem, the sandwich theorem,
14. The Leray-Schauder degree: construction (contraction property for the Brouwer degree), properties, nonlinear alternative
Presentations.
15. Bifurcation theory: (a) Morse theory, (b) theorems by Krasnosielski and Rabinovitz
16. Convex sets and their extreme points: the Krein-Millman theorem and applications
17. The semigroup approach to first order quasilinear equations in several space variables.
18. Isometrical embeddings with positive Gauss curvature: proof of openess in the continuity method
19. The Krein-Rutman theorem
20. Young measures: existence theorem and basic properties
21. Optimal mass transportation: the Monge-Kantorovich problem
22. Generalized implicit function theorems: the Nash-Moser theorem
23. Helmholtz decomposition and Stokes operator

Calendar.

5 Sept (W): First class
29 Oct (M): First Midterm
23 Nov (F): Thanksgiving - no class
10 Dec (M): Second Midterm
12 Dec (W): Last class

23 Jan (W): First class
22 Feb (F): No class
12 March - 21 March (M, W, F): Spring break - no classes
9 May (F): Last class