Course syllabus
Course PM
This page contains the program of the course: lectures, exercise sessions, assignments. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM.
Program
The schedule of the course is in TimeEdit.
Course literature: M. Einsiedler and T. Ward, Functional Analysis, Spectral Theory and Applications. Springer GTM.
Chapt. 2, 2.1—2.2. 2.4. (Section 2.2 on space of continuous functions and Stone - Weierstrass theorem is covered by the course Real Analysis, MMA120, by Ulla Dinger).
Chapt. 3, 3.1-3.2.
Chapt. 4, 4.1-4.2.
Chapt. 5, 5,1.
Chapt. 7, 7.1-7.4.
Chapt. 8. 4.1 (Banach-Alaoglu theorem), 4.2 (Applications of B-A theorem, part of)
Brief summary:
The course has roughly two parts, the "abstract" Functional Analysis, and the "concrete" Functional Analysis. The "abstract" part is to develop general theory for Banach spaces and operators, and it is in
Chapt. 2.1-2.2, Chapt 3.1, Chapt 4.
The "concrete" part is on the specialization and application of the abstract theory to concrete spaces and it is in
Chapt 2.3, 3.1.3, Chapt 7.
The plan of the course is that we cover the "abstract Functional Analysi" first and then the "concret Functional Analysis".
List of Theorems whose statements and proofs are required for the exam.:
- Proposition 2.35 (non-compactness)
- Theorem 3.13 (for convex sets in Hilbert spaces)
- Corollary 3.19. (Riesz representation)
- Theorem 4.1 (Banach-Steinhauss)
- Theorem 4.12 (Baire category)
- Theorem 4.28 (Closed graph)
- Hahn-Banach
- Existence of Banach limit on l ^ \infty (small l infinity)-
- Banach-Alaoglu Theorem
- Dual space of L^1.
Lectures (with a summary of completed sections)
Tuesdays and Thursdays classes will be lectures, and Fridays will be exercise classes and discussions.
Supplementary text/solutions to exercises:
(1) "Elementary" proof of Prop. 2.6 without using compactness
(2) Elementary proof of Fredholm index is invariant under finte rank pertubations
(3) Weak (weak*)-closure of the sphere.
| Week |
Content |
Sections [EW] | Summary |
|---|---|---|---|
| 45 |
Normed vector spaces. Quotient of normed vector spaces |
2.1.1-2.2.2 |
Definitions of norms. Proof of l^p-norm is a norm. Hölder ineq. Criterior for quotient norm being a norm |
| 46 |
Non-compactness of unit ball in infinite-dimensional Banach spaces. Bounded operators and linear functionals. (We leave 2.3.3 to later parts; the abstract Stone-Weirstrass theorem is not part of the course - we might have time later to prove it.) |
2.2.3-2.4.1 | Proposition 2.35. Motivation and Definition of uniform convexity. Techniques for computing quotient norms. |
| 47 | Banach algebras. Hilbert spaces and generally uniformly convex spaces (such as l^p spaces for p in [2, \infty) |
2.4.2, 3.1.1-3.1.2, 4.1 (Banach-Steinhaus) |
B-S theorem on uniform boundedness of family of operators. Reformulation in terms of union and intersections inverse images of balls. |
| 48 | Baire Theorem. Open mapping theorem (OMT). Closed Graph Theorem. | 4.1,1-4.2.3. | Finished the proof three major theorems of 4 theorems in Linear Functional Analysis: Banach-Steinhuas Thm, OMT and CGT. The OMT might be the most intuitive one whereas CGT is the most useful one. |
| 49 |
Hahn-Banach theorem. Compactness results: Banach-Alaoglu for the closed unit ball in dual space X^*. |
2.3.1. 7.1.1. |
Hahn-Banach theorem and applications, Banach limit. w*-topology on X^*. Examples of complemented (i.e. split) subspaces, non-complementedness of c_o in l^\finty (see also the file here) |
| 50 | Dual spaces of L^p. (l^p dual was computed in week 47). Hölder inequality (integral version) | 7.1.1-7.2.2, 7.3-7.4 |
Paring of (f, g) for f\in L^p and g\in L^q and maximizer. Radon-Nikydym derivatives (to obtained functions from measures). Riesz Representation for C(X). |
| 51 | Arzela-Ascoli. Applications. Equi-distributions. | 2.3.3, 8.2.1, 7.2.3 | Invariant measure. Ergodic measure with respect to a transformation. Application of B-Alaoglu them to equi-distributions and convergence of measures. |
| v3 | Examination (obs! week 3) |
Recommended exercises
(There are so called Essential Exercises in the book - please try to work them out. Exercises marked with * might be a bit difficult. Some extra exercises closely related to the lectures will be handed out together with the assignments.)
2.3a), 2.7, 2.9, 2.16, 2.18, 2.25, 2.26, 2.36*, 2.55, 2.56, 2.58
3.4, 3.5, 3.9, 3.10, 3.14 (1)(2), 3. 15 (Do this for R^n with l^p-norm first). 3.20, 3.27, 3.28*, 3.37. 3.40
4.4, 4.5, 4.16, 4.18, 4.20, 4.23, 4.29. 7.2, 7.7, 7.11, 7.33 (a)-(d),
7.56, 7.58. 8.6, 8.8, 8.9, 8.12*, 8.16, 8.17, 8.18.
Demonstration Exercises:
Week 45: Different elementary proof of equivalence of norms on R^n. Examples of unbounded linear functionals. (See Assignment-1 with extra exerecises).
Week 46: Exercise 2.16 (please make some preparation - here is some warm-up exercise and generalizations). Extra-Ex 7, 8. (See the file Assignment 1 in pd).
Week 47: Exercise 2.36* (This exercise is a bit difficult. Please prepare and try it in advance. Try to read the next section on Arrzela-Ascoli theorem ). Exercise 3.15*
Week 48: Exercise 4.4. 4.20. Some exercises in Assignment 2.
Week 49: 3.28 (on c_0 being non-complemented subspace of l^\infty.) (My attempt using Banach algebra does not work. It is a rather hard problem - see the file here). Assignment and Extra-Exercise Part 3.
Week 50: Ex. 7.2, 7.7. (These are some kind of "standard" exercises.) Exercise 7.11 (see also the extra exercise 9 in assighment 1)
Week 51: Ex 8.8 (This is also some standard ex. Try to work it out on your own). Ex. 8.16, 8.18. (Solution of Ex. 8.18)
Assignments
(There will be 5 homework assignments, each will be graded with points and will be added up along with the oral examination to the final grade.)
On exercise 1: You can assume that N(x) is a smooth function and the Hessian of N(x) is semi-positive definite (if you're not very familar with differentiation of non-smooth functions). This exercise is to confirm our intuition that |x|^c is convex for c>1.
Course summary:
| Date | Details | Due |
|---|---|---|