MATH 418 - Honors Real Variables II
(Fall 2021)
MATH 516 - Linear Analysis (Fall 2021)
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 3:00 - 5:00 pm or by appointment.
General information
NOTE: All information on this site also is
available from the official course site on eClass.
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
In class, we are going to work with a set of fairly
detailed notes which will be posted on eClass over the
course of the semester. You may want to give the relevant section(s) of the notes at least a cursory read before coming to class.
In addition, be prepared to take
careful notes in class, as needed.
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
1
Sep
|
Preliminaries: linear algebra; metric topology.
|
Preliminaries.
|
| 2
|
Fri
3
Sep
|
Preliminaries: metric topology, completeness and
completion(s), compactness, convexity.
|
|
|
|
Mon
6 Sep
|
No class.
|
Happy Labour Day! |
| 3
|
Wed
8
Sep
|
Basic properties of norms. Banach spaces.
|
Chapter 1.
|
| 4
|
Fri
10
Sep
|
First examples. Completing a normed space.
|
|
| 5
|
Mon
13
Sep
|
Series in normed spaces: (absolute; unconditional)
convergence; summability. Quotients and products.
|
|
| 6
|
Wed
15
Sep
|
More examples. Finite- vs. infinite-dimensional spaces. Equivalent norms.
|
|
| 7
|
Fri
17
Sep
|
Riesz lemma. Characterizations of
finite-dimensionality. Examples.
|
(class on zoom)
|
| 8
|
Mon
20
Sep
|
Linear operator basics.
|
|
| 9
|
Wed
22
Sep
|
Examples of operator norms.
|
|
| 10
|
Fri
24
Sep
|
(Isometrically) isomorphic spaces and equivalent
operators. Compact operators.
|
|
| 11
|
Mon
27
Sep
|
Dual spaces and operators. Examples.
|
|
| 12
|
Wed
29
Sep
|
Extending linear functionals.
|
|
| 13
|
Fri
1
Oct
|
The Hahn-Banach Theorem.
|
|
| 14
|
Mon
4
Oct
|
The bidual. Reflexive spaces.
|
|
| 15
|
Wed
6
Oct
|
The Uniform Boundedness Theorem.
|
|
| 16
|
Fri
8
Oct
|
The Open Mapping and Closed Graph Theorems.
|
|
|
|
Mon
11 Oct
|
No class.
|
Happy Thanksgiving! |
| 17
|
Wed
13
Oct
|
Applications: The dual of C[0,1]; regular summation schemes.
|
|
| 18
|
Fri
15
Oct
|
Complemented subspaces.
|
Practice midterm I posted on eClass.
|
| 19
|
Mon
18
Oct
|
Semi-inner products. Cauchy-Schwarz inequality. Hilbert spaces.
|
Chapter 2.
|
| 20
|
Wed
20
Oct
|
Basic properties of inner product spaces.
|
|
|
|
Fri
22
Oct
|
MIDTERM TEST 1.
|
Good luck!!
Model solutions posted on eClass.
|
| 21
|
Mon
25
Oct
|
Orthogonal complements and projections. The Fréchet-Riesz theorem.
|
|
| 22
|
Wed
27
Oct
|
Orthonormal systems and bases.
|
|
| 23
|
Fri
29
Oct
|
Gram-Schmidt procedure. Examples of orthonormal bases.
|
|
| 24
|
Mon
1
Nov
|
Fejér's Theorem and some consequences. Orthonormal bases in L2[0,1].
|
|
| 25
|
Wed
3
Nov
|
Classical Fourier series.
|
|
| 26
|
Fri
5
Nov
|
Linear operators on Hilbert spaces. The adjoint operator.
|
|
|
|
|
|
Enjoy Reading Week. |
| 27
|
Mon
15
Nov
|
Properties of normal operators.
|
|
| 28
|
Wed
17
Nov
|
Positive operators. Projections.
|
|
| 29
|
Fri
19
Nov
|
Invariant subspaces. Introducing the spectrum of an operator.
|
Chapter 3.
|
| 30
|
Mon
22
Nov
|
Examples and properties of spectra.
|
|
| 31
|
Wed
24
Nov
|
Further properties of spectra.
|
|
|
|
Fri
26
Nov
|
MIDTERM TEST 2.
|
Good luck!!
Model solutions posted on eClass.
|
|
|
|
|
Please consider
completing the online MATH 418/516 course survey. It only takes a
few minutes. Thank you. |
| 32
|
Mon
29
Nov
|
Compact operators and their spectra.
|
|
| 33
|
Wed
1
Dec
|
Normal operators and their spectra.
|
|
| 34
|
Fri
3
Dec
|
The spectral theorem for (compact) normal operators.
|
|
| 35
|
Mon
6
Dec
|
Concluding remarks.
Final housekeeping.
|
|
|
|
|
|
|
|
|
|
|
Good bye and good
luck !! |
|
|
|
|
|
|
|
Mon
13 Dec
|
|
Pre-exam "open house":
10am - 3pm in CAB 683 and on zoom - please use the sign-up
sheet on eClass.
|
|
|
|
|
|
|
|
Mon
20 Dec
|
|
Pre-exam "open house":
10am - 3pm in CAB 683 and on zoom - please use the sign-up
sheet on eClass.
|
|
|
|
|
|
|
|
|
|
|
|
|
Tue
21 Dec
|
Final exam !
2-4pm in CAB 269
Please see information on eClass for details.
|
Good luck !! |
|
|
|
|
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|
