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 #
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Date
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Material covered / special events
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Remarks/ additional material
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WELCOME TO MATH 418/516 !!
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1
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Wed 1 Sep
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Preliminaries: linear algebra; metric topology.
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Preliminaries.
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2
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Fri 3 Sep
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Preliminaries: metric topology, completeness and
completion(s), compactness, convexity.
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Mon 6 Sep
|
No class.
|
Happy Labour Day! |
3
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Wed 8 Sep
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Basic properties of norms. Banach spaces.
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Chapter 1.
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4
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Fri 10 Sep
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First examples. Completing a normed space.
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5
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Mon 13 Sep
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Series in normed spaces: (absolute; unconditional)
convergence; summability. Quotients and products.
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6
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Wed 15 Sep
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More examples. Finite- vs. infinite-dimensional spaces. Equivalent norms.
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7
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Fri 17 Sep
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Riesz lemma. Characterizations of
finite-dimensionality. Examples.
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(class on zoom)
|
8
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Mon 20 Sep
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Linear operator basics.
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9
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Wed 22 Sep
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Examples of operator norms.
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10
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Fri 24 Sep
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(Isometrically) isomorphic spaces and equivalent
operators. Compact operators.
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11
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Mon 27 Sep
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Dual spaces and operators. Examples.
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12
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Wed 29 Sep
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Extending linear functionals.
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13
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Fri 1 Oct
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The Hahn-Banach Theorem.
|
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14
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Mon 4 Oct
|
The bidual. Reflexive spaces.
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15
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Wed 6 Oct
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The Uniform Boundedness Theorem.
|
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16
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Fri 8 Oct
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The Open Mapping and Closed Graph Theorems.
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|
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Mon 11 Oct
|
No class.
|
Happy Thanksgiving! |
17
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Wed 13 Oct
|
Applications: The dual of C[0,1]; regular summation schemes.
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18
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Fri 15 Oct
|
Complemented subspaces.
|
Practice midterm I posted on eClass.
|
19
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Mon 18 Oct
|
Semi-inner products. Cauchy-Schwarz inequality. Hilbert spaces.
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Chapter 2.
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20
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Wed 20 Oct
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Basic properties of inner product spaces.
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|
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Fri 22 Oct
|
MIDTERM TEST 1.
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Good luck!!
Model solutions posted on eClass.
|
21
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Mon 25 Oct
|
Orthogonal complements and projections. The Fréchet-Riesz theorem.
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22
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Wed 27 Oct
|
Orthonormal systems and bases.
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23
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Fri 29 Oct
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Gram-Schmidt procedure. Examples of orthonormal bases.
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24
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Mon 1 Nov
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Fejér's Theorem and some consequences. Orthonormal bases in L2[0,1].
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25
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Wed 3 Nov
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Classical Fourier series.
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26
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Fri 5 Nov
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Linear operators on Hilbert spaces. The adjoint operator.
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|
|
|
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Enjoy Reading Week. |
27
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Mon 15 Nov
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Properties of normal operators.
|
|
28
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Wed 17 Nov
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Positive operators. Projections.
|
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29
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Fri 19 Nov
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Invariant subspaces. Introducing the spectrum of an operator.
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Chapter 3.
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30
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Mon 22 Nov
|
Examples and properties of spectra.
|
|
31
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Wed 24 Nov
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Further properties of spectra.
|
|
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Fri 26 Nov
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MIDTERM TEST 2.
|
Good luck!!
Model solutions posted on eClass.
|
|
|
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Please consider
completing the online MATH 418/516 course survey. It only takes a
few minutes. Thank you. |
32
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Mon 29 Nov
|
Compact operators and their spectra.
|
|
33
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Wed 1 Dec
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Normal operators and their spectra.
|
|
34
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Fri 3 Dec
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The spectral theorem for (compact) normal operators.
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35
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Mon 6 Dec
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Concluding remarks. Final housekeeping.
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Good bye and good
luck !! |
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Mon 13 Dec
|
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Pre-exam "open house":
10am - 3pm in CAB 683 and on zoom - please use the sign-up
sheet on eClass.
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Mon 20 Dec
|
|
Pre-exam "open house":
10am - 3pm in CAB 683 and on zoom - please use the sign-up
sheet on eClass.
|
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Tue 21 Dec
|
Final exam !
2-4pm in CAB 269
Please see information on eClass for details.
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Good luck !! |
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