MATH 418 - Honors Real Variables II (Fall 2019) MATH 516 - Linear Analysis (Fall 2019)

Time and Location

Time: MWF 10:00 - 10:50 am
Room: CAB 365

Instructor

Dr. Arno Berger (CAB 683, berger@ualberta.ca)

Office hours

MWF 2:00 - 4:00 pm or by appointment.

General information

Please see this PDF document for all relevant details concerning MATH 418/516. (An abbreviated version of this document will be distributed in class.)

Course notes

Be prepared to take careful notes in class, as no set textbook will be used.

You are welcome to use the fine notes by Dr. Runde (PDF, 0.7MB) for background reading. Be aware, however, that notation and terminology in these notes may differ from the ones used in class.

Material covered in class (Course Diary)

I plan to keep an up-to-date list of the topics, examples etc. covered in class.

Lecture # Date Material covered / special events Remarks/ additional material
WELCOME TO
MATH 418/516 !!

1 Wed
4
Sep
Preliminaries: linear algebra; the contraction mapping theorem. Handout: Preliminaries.
2 Fri
6
Sep
Preliminaries: metric topology, completeness and completion(s), compactness.
3 Mon
9
Sep
Convex sets and functions. Normed spaces.
4 Wed
11
Sep
Basic properties of norms. Banach spaces.
5 Fri
13
Sep
First examples. Homework #1 posted on eClass - due 25 September.
6 Mon
16
Sep
Series in normed spaces: (absolute; unconditional) convergence; summability.
7 Wed
18
Sep
Quotients and products. More examples.
8 Fri
20
Sep
L^p spaces - a very brief review.
9 Mon
23
Sep
Finite- vs. infinite-dimensional spaces. Equivalent norms.
10 Wed
25
Sep
Riesz lemma. Characterizations of finite-dimensionality. Examples.
11 Fri
27
Sep
Schauder bases. Linear operator basics. Homework #2 posted on eClass - due 16 October.
12 Mon
30
Sep
Spaces of linear operators.
13 Wed
2
Oct
Examples. (Isometrically) isomorphic spaces.
14 Fri
4
Oct
Banach-Mazur distance. Dual spaces.
15 Mon
7
Oct
Dual operators. Examples.
16 Wed
9
Oct
Extending linear functionals.
17 Fri
11
Oct
The Hahn-Banach Theorem.
Mon
14 Oct
No class. Happy Thanksgiving!
18 Wed
16
Oct
The bidual. Reflexive spaces. The uniform boundedness theorem. Homework #2 deadline extended to 4pm on 18 October.
19 Fri
18
Oct
The Banach-Steinhaus and open mapping theorems. Homework #3 posted on eClass - due 6 November.

Practice midterm I posted on eClass.
20 Mon
21
Oct
The closed graph theorem. Applications.
Wed
23
Oct
MIDTERM TEST 1. Good luck!!

Model solutions posted on eClass.
21 Fri
25
Oct
The dual of C[0,1].
22 Mon
28
Oct
Complemented subspaces. Consistent summation schemes.
23 Wed
30
Oct
Semi-inner products.
24 Fri
1
Nov
The Cauchy-Schwarz inequality and the parallelogram law.
25 Mon
4
Nov
Examples. Orthogonality and the closest point property.
26 Wed
6
Nov
Orthogonal complements and projections. Homework #4 posted on eClass - due 20 November.

Graduate project posted on eClass - due 6 December (last day of class).
27 Fri
8
Nov
The Fréchet-Riesz theorem. Orthonormal systems.
28 Mon
18
Nov
Orthonormal bases. Examples.
29 Wed
20
Nov
Orthonormal bases in L2[0,1]. Homework #5 posted on eClass - due 6 December.

Practice midterm II posted on eClass.
30 Fri
22
Nov
Fejér's Theorem and some consequences.
31 Mon
25
Nov
Classical Fourier series.
Wed
27
Nov
MIDTERM TEST 2. Good luck!!

Model solutions posted on eClass.
32 Fri
29
Nov
Linear operators on Hilbert spaces.
Please consider completing the online MATH 418/516 course survey. It only takes a few minutes. Thank you.
33 Mon
2
Dec
The spectrum. Definitions and examples.
34 Wed
4
Dec
Properties of the spectrum.
35 Fri
6
Dec
Concluding remarks.

Good bye and good luck !!
Fri
13 Dec
office hours (also for MATH 201):
10am - 3pm in CAB 683.
Mon
16 Dec
office hours:
2pm - 5pm in CAB 683.
Tue
17
Dec
Final exam !

Please see box on the left for details.
Good luck !!

Homework

Throughout the semester, a total of five assignments will be posted on eClass. Unless stated otherwise, the deadline for homework submission is 4:00 pm on Wednesday. Please submit your solutions into the designated MATH 418/516 assignment box on the third floor of CAB.

Midterm test

Two midterm tests will be held on Friday October 23 and November 27, 2019, respectively, in class.

Some details about the first midterm:

• Duration: 50 minutes.
• Material covered: up to, and including Example I.60, i.e., the Hahn-Banach theorem and some of its consequences.
• NO textbooks, notes, calculators, formula sheets etc.!
• NO cell-phones, laptops, or other electronics!
• Please bring a valid ID with you.
• Good luck!

To help you prepare for the midterm test, a practice version has been posted on eClass - the real test will be very similar.

Some details about the second midterm:

• Material covered: up to, and including Corollary II.29, with an emphasis on inner product spaces.
• All other details (50 min duration, no books or electronic aides, etc.) are exactly as on the first midterm.
• Good luck!

To help you prepare for this test, a practice version has been posted on eClass - just as on midterm I, the real test will be very similar.

Final exam

The final exam will be held on Tuesday, December 17, 2016 at 2:00 pm, in CAB 281.

• Duration: 2 hours.
• Material covered: Basically everything, but with more emphasis on Chapters II and III.
• NO textbooks, notes, calculators, formula sheets etc.!
• NO cell-phones, i-pods, or other electronics!
• Please bring a valid ID with you.
• Good luck!

To help you prepare for the final, a practice version has been posted on eClass. The real exam will be very similar.

Other material

General information about the department's honors program is available here.