Funktionalanalys II (5hp), HT 2008
Kurskod 1MA044.
The exam is corrected
Results for those wanting results by internet.
The exams can be obtained at my office any afternoon this week.
Kursplan.
Kursen bygger på
Funktionalanalys I.
Lärare
Kursstart 2 september 13.15.
Kurslitteratur
- Peter Lax: Functional Analysis.
Wiley, New York, 2002.
-
W. Rudin:
Functional Analysis.
2nd ed.
McGraw-Hill, New York, 1991
En kort engelsk-svensk ordlista finns nedan.
Homework
Some homework assignments will be given.
These form part of the examination.
Correct solutions will give bonus points on the exam
(up to 3 bonus points per set, i.e., total maximum 9).
Solutions can be given to me at lectures, or in my mailbox at
Ångström (floor 4), at the latest on the days shown below.
- 15 September.
- 29 September.
- 27 October.
Lectures
Here is a rough list of contents for the lectures, with approximate
indications of sections in the textbooks (denoted L and R).
- 2/9: Topological vector spaces, locally convex spaces,
seminorms. L 13.0, 3.2; R 1.1-1.7, 1.33-1.37.
- 4/9: Locally convex spaces, seminorms, Hahn-Banach theorem.
L 3.1, 3.2, 13.1; R 1.14, 1.38, 3.1-3.4, 3.11, 3.14
- 9/9: Banach algebras. L 17.1. (R 10.1-3, 10.10-12)
- 10/9: Integration of vector-valued functions. Vector-valued
analytic functions. Resolvent, spectrum, spectral radius.
L 11.4, 17.1, Theorem 18.12 (R 3.26, 3.30-3.32, 10.13-14)
- 11/9: Functional calculus in a Banach algebra (L 17.2)
- 15/9: Bounded self-adjoint operators in a Hilbert space: spectrum and
functional calculus L 31.1-2. (R 12.7-12.11)
- 16/9: Functional calculus, spectral resolution L 31.3, (R 12.17,
12.21, 12.23-24).
- 17/9: positive operators, polar decomposition, normal operators,
unitary operators, unbounded self-adjoint operators, spectral
representation L 31.5-7, 32.1; R 12.13-15, 25-26, 32-35, 13.30
L 31.5,6,7
- 18/9: PROBLEM SOLVING: Homework 1; R Exercises 12: 9, 10, 19,
20, 26, 36, 37, L Exercises Ch. 31.
- 29/9: Hilbert-Schmidt operators, singular values L 30.8, 30.1
- 1/10: Trace class operators, trace. L 30.2, 30.5
- 2/10:
Applications to differential equations L 7.2, 29.2.
PROBLEM SOLVING: Homework 2;
L Exercises Ch. 30: 1, 2, 3, 4, 5, 6, 11(d)(h)(i)(j)
- 7/10: Sobolev spaces. L p. 40 (and more)
- 8/10: The Sobolev embedding theorem. The Riesz-Thorin interpolation
theorem. L 16.2
- 9/10: The Marcinkiewicz interpolation theorem.
Exam
Check
exam schedule
for changes.
En kort ordlista
Detta är ett litet komplement till
Anders Vretblads allmänna
matematiska ordlista.
| Engelska
| Svenska
|
| accumulation point
| hopningspunkt
|
| adjoint operator
| adjungerad operator (adjunkt)
|
| Banach space
| Banachrum
|
| bounded
| begränsad
|
| closed
| sluten
|
| closure
| slutna höljet
|
| complete
| fullständig
|
| domain
| definitionsområde
|
| dual, dual space
| dual, dualt rum
|
| functional
| funktional
|
| Hilbert space
| Hilbertrum
|
| inner product
| inre produkt
|
| inner product space
| inreproduktrum
|
| kernel
| kärna
|
| linear space
| linjärt rum
|
| metric
| metrik
|
| metric space
| metriskt rum
|
| norm
| norm
|
| normed space
| normerat rum
|
| null space
| nollrum
|
| open
| öppen
|
| orthogonal
| ortogonal
|
| projection
| projektion
|
| range
| bild, bildrum
|
| space
| rum
|
| total
| total
|
| trace
| spår
|
| trace class
| spårklass
|
| unbounded
| obegränsad
|
| vector space
| vektorrum
|
På svenska skrivs Banachrum, Hilbertrum
och andra sammansättningar med personnamn som ett ord
(liten begynnelsebokstav går också bra).