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 (under H21/Pages)
•  Schedule, Week 44-Week 50 (Nov. 1- Dec. 18): Tuesday 10:00-12:00,Thursday 13:15 - 15:00, Friday 08:00-10:00. MVH11.
• The oral examination will be in January 2022 and we shall fix the schedule.

 

 

Program

Schedule/Textbook/Content/Plan.

• The schedule of the course is in TimeEdit.
• Course literature:  [EW] M. Einsiedler and T. Ward, Functional Analysis, Spectral Theory and Applications. Springer GTM. 
• Content: [EW] 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.
• Overall plan: The plan of the course is that we  cover the "abstract Functional Analysi" first, the general properties of Banach spaces and operators,  and then  the "concret Functional Analysis" with applications such as L^p-spaces, Ascoli-Arzela theorem and equi-distributions.

           

List of Theorems whose statements and proofs are required for the exam.:

1. Proposition 2.35 (non-compactness)
2. Theorem 3.13 (for convex sets in Hilbert spaces; you are welcome to  present the general result and the proof for uniformly convex Banach spaces.)
3. Corollary 3.19. (Riesz representation)
4. Theorem 4.1 (Banach-Steinhauss)
5. Theorem 4.12 (Baire category)
6. Theorem 4.28 (Closed graph)
7. Hahn-Banach Lemma and Theorem
8. Existence of Banach limit on l ^ \infty (small l infinity)-
9. Banach-Alaoglu Theorem
10. Dual space of L^1.

Examples of  Exam Questions: Examples

     

Lectures (with a summary of completed sections)

Tuesdays and Thursdays classes will be lectures, and Fridays will be exercise classes and discussions. (Last years' Zoom-lecture notes can be found in the Canvas/MMA120 H20. This year we have been able - so far till now Nov 28 - to cover more challenging exercises on Fridays due to the possibility of class room discussions and interests.)

      

 

Week	Content	Sections [EW]	Summary
44	Introduction/Motivation. Normed vector spaces. Quotient of normed vector spaces	2.1.1-2.2.2	Criteria for quotient semi-norms being a norm. Concrete computation of quotient norms, dual norms for finite-dimensional normed spaces.
45	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 (Section 2.3.1 on Arzela-Ascoli theorem is moved to Week 48, it will be treated with other related compactness results.) 2.4.2, 3.1.1-3.1.2.	Hilbert spaces and uniform convexity and unique approximations.
46	Banach-Steinhauss/Baire/Open Mapping Theorems.	4.1,1-4.2.1-2.	B.S. and its generalization to Baire Thm. Application of Baire Thm to Open Mapping Thm.
47	Closed Graph Theorems (CG). Hahn-Banach theorem (H-B). Banach-Limits and applications.	4.2.3.  2.3.1.  7.1-7.2	H-B Lemma and Theorem and applications: Separation of points/sets by linear functional, extension of the concept of limits.
48	Compactness results: Banach-Alaoglu for the closed unit ball in dual space X^*. Arzela-Ascoli.	8.1.  2.3.1	Duals and pre-duals; weak* and weak topology. Dual and pre-dual operations via the pairing X\times X^*\to R for subspaces (and convex closed subsets) using the trivial subspace 0 of R (and [-1, 1]\subset R).
49	Dual spaces of L^p. (l^p dual was computed  in week 47). Hölder inequality. Applications.	7.3 - 7.4	Riesz Representation Theory for L^p and for positive functional on C(X). Relation between L^p-spaces for different p's.
50	Applications. Equi-distributions.	2.3.3 (2.3.2), 8.2.1 (7.2.3)	Equi-distributed sequences on [0, 1]/Circle, [0, 1]^2/Torus. Invariant measures, ergodic measures vs extreme points and application to equi-distributions.
W2	Examination (Week 2, Thursday and Friday)

 

 

 

 

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 together with assignments exercises will be given - see Canvas/Assignments.)

     2.3a), 2.7, 2.9, 2.16, 2.18, 2.25, 2.26, 2.36*, 2.55, 2.56, 2.58. 2.50 (Equi-distribution)

     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 44: Norms/Examples/Quotient Norms.  (a) Examples of norms on matrices (Extra Ex. 4). (b) Computation of quotient norms (Extra Ex. 14, a simpler version). (c) Dual norms and convex polygons (Extra Ex. 15.)

     Week 45:  Understanding concrete norms and uniform convexity.  [EW] Ex. 2.16 and Extra Ex. 20. l^p-norm and its Hessian. Ex. 3.15

     Week 46:   Application of B.S and G-delta Sets. Exercises in Assignment 1. Construction of continuous functions with divergent Fourier series (see [EW] pp.125-126, and read again the proof of Theorems 4.1, 4.9). [EW] Ex. 4.20.

     Week 47:  Hahn-Banach theorem and Separations of sets by functionals.   [EW] Ex. 7.2, 7.7. (These are some kind of "standard" exercises.)   Exercise 7.11.  Mazur theorem on separating points from convex closed balanced subsets.

     Week 48: Complemented subspaces.  Weak/Weak*-toplogy.   [EW] Ex. 3.28 (on c_0 being non-complemented  subspace of l^\infty). This is also a bit difficult, please read this in advance. Discussions on compact subsets in l^\infty. Ex. Ex 8.8 (This is standard ex. Try to work it out on your own first and/or  discuss  during the exercise class). 

     Week 49:. Dual space of L^p/l^p and C(X), Hölder inequality (which implies the convexity of l^p-norm giving a different proof),   generalized Hölder inequality for 1/p +1/q=1/r. Appendix B, Theorem B. 15. Ex. 7.33, Questions on  Assignments 2, 3. Extra Exercise 43 (see Assgnment 5)  on L^p-spaces.

     Week 50:  Equi-distributions. [EW] Ex. 2.48, Ex 2.50 (or its variations). Extra Ex. 51 (see the Assignment  file). Questions in Assignment 4 and 5.

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. They are in Canvas/Assignments where you can also upload your solutions - you may of course also hand over your written solutions.

 

Student representatives

The following students has been appointed:

 

Victor Ahlquist, gusahlquvi@student.gu.se

 

All critiques or feedbacks are welcome, to me or via Victor.

 

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