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.
•  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 2023 and we shall fix the schedule.

 

Program

P1. Schedule.

• The schedule of the course is in TimeEdit.

P2. Litterature/Content.

• 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.

           

P3. Theory Requirements.

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 may of course present a general proof for uniformly convex Banach spaces if you can.)
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. Lemma 7.1 and Theorem 7.3 (Hahn-Banach Lemma and Theorem)
8. Corollary 7.14 (Existence of Banach limit on l^\infinity(N)).
9. Theorem 8.10 (Banach-Alaoglu Theorem)
10. Proposition 7.36 (Dual space of L^p, p >1 for finite measure space.)

P4. Lectures and Exercises.

Tuesdays and Thursdays classes will be lectures, and Fridays will be exercise classes, discussions and/or lectures.

 

Lectures.

  

Week	Content	Sections [EW]	Important Theorems
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.
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.	Compactness. Examples of compact and non-compact subsets in Banach spaces. Hilbert spaces.
46	Banach-Steinhauss/Baire Category Thm	3.1.2. 4.1.1-4.2.2.	Uniform convexity and unique approximations.B.S. and its generalization to Baire Thm. Application of Baire Thm.
47	Open Mapping Theorem. Closed Graph Theorem. Existence of Bounded Inverse. Introduction to Hahn-Banach Lemma.	4.2.3. 7.1.1.	Direct sum of Banach spaces. Relation between an operator and its graph in the direct sum; natural inclusion and projection. Hahn-Banach lemma and projection on one-dimensional subspaces.
48	Hahn-Banach Lemma/Theorem. Banach Limit. Amenable Groups. Weak/Weak* topology. Banach-Alaoglu Thm on *-compactness	7.1.1-7.2.2. 8.1	Existence of linear functional with special property (maximizers, separations, invariance). Compactness.  Dual spaces of L^p.
49	Dual spaces concrete spaces:  (L^p, L^q)-paring, Riesz theorem for C(X). Arzela-Ascoli Theorem for norm-compact subsets in C(X). Weak-topology in C(X) and weak-* topology in the dual space.	7.3, 7.4. 2.3.	Linear functionals on C(X)  as measures on X and related topologies in C(X) and C(X)*
50	The space C(X), its dual space and application to equip-distributions for points in X.	2.3.1-2.3.3. 8.2.	Continuation of weak*-compactness and application to sequences of measures.
W2/W3, 2023	Oral examination

 

 

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 sequences and matrices, l^p-norms and Schatten-von Neumann norm (Extra Ex. 5). (b) Computation of quotient norms (Extra Ex. 15, a simpler version).

     Week 45:  Understanding concrete norms and uniform convexity.   Dual norms and convex polygons (Extra Ex. 16.  [EW] Ex. 2.16 and Extra Ex. 20 on comparing different l^p-spaces.   Ex. 2.36* (compact subsets in l^p; discussions and hints).

     Week 46:  Uniform Convexity and B-S.  [EW] Ex. 3.15 (uniform convexity of l^p-norms),  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 on G-delta set.

     Week 47: Boundedness/Invertible vs Closedness/Openess. [EW] Ex 4.29. Exercises in Assignment 2 on complemented subspaces (see also [EW] Ex. 3.28 on non-complemented subspaces - which we'll discuss during the exercise classes.). Extra Ex. 46/47 (See Assignment 3.)

     Week 48: Hahn-Banach theorem and Mazur's theorem on separations of sets by functionals.   Complemented subspaces.  Weak/Weak*-topology.  We'll find some hints to Assignment 4 which is a bit difficult. [EW] Ex. 7.2, 7.7. (These are some kind of "standard" exercises.)   Exercise 7.11.   [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.

     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 Exercises 42/43  on L^p-spaces (see the file Assignment 4).

     Week 50:  Equi-distributions. [EW] Ex. 2.48, hints to Ex 2.50 and the Assignment 5. Extra Ex. 51 on examples of equi-distributions. Questions in Assignment 4 and 5. Review and Summary of Exercises for the whole course.

 

Examination.

The examination consists of

(A) 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.

 

(OE) Oral Examination

Examples of  Exam Questions: Examples

 

Course Representative(s)/Course Evaluations. (See also the course PM)

 

All critiques/feedback are welcome. You may contact the course representative(s) (see course PM) and me. There will be a evaluation polling sent out in the end of the course and you're encouraged to fill the form.

 

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