Funktionalanalys II (5hp), HT 2010
Kurskod 1MA044.
Kursplan.
Kursstart 1 september 10.15.
Lärare
Schema
Schema
Denna vecka
Kurslitteratur
- Peter Lax: Functional Analysis.
Wiley, New York, 2002.
-
W. Rudin:
Functional Analysis.
2nd ed.
McGraw-Hill, New York, 1991
Rudin är slut på förlaget (men kan köpas t.ex. på www.amazon.co.uk).
Jag kommer därför att använda Lax som huvudbok och bara använda Rudin för
vissa kompletteringar.
En kort engelsk-svensk ordlista finns nedan.
Homework
Three 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.
- 16 September.
- 6 October (changed).
-
12 October.
Lectures
Here is a rough list of contents for the lectures, with approximate
indications of sections in the textbook (L = Lax; R = Rudin).
- 1/9: seminorms, Hahn-Banach theorem.
L 3.1, 3.2, Theorem 3.8.
- 3/9: Topological vector spaces, locally convex spaces,
seminorms. L 13.0 + R 1.1-1.7.
- 7/9: Locally convex spaces and seminorms. R 1.33-1.38.
- 9/9: Weak and strong topologies. Vector-valued analytic
functions. Hahn-Banach again. Banach algebras.
L 11.4, 12, 17.1 + R 3.4-3.6.
- 13/9: Resolvent, spectrum, spectral radius. L 17.1
- 15/9: Gelfand-Mazur theorem. Functional calculus in a Banach algebra.
L 17.2, Theorem 18.12.
- 16/9: Bounded self-adjoint operators in a Hilbert space. Functional
calculus. Positive operators. Polar decomposition. L 31.1-2.
- 22/9: PROBLEM SOLVING: Homework 1;
L Exercise 17.2, 17.3, 17.4; R Exercise 10.1, 10.4, 10.7.
- 23/9: Spectral resolution, spectral representation. L 31.3, 31.5 (partly).
(R 12.17, 12.21, 12.23-24).
- 24/9: Point spectrum and eigenvalues.
Normal operators,
unitary operators.
Unbounded self-adjoint operators.
L 31.6-7, 32.1 (R 12.11-15, 12.25-26, 13.30)
- 28/9: Hilbert-Schmidt operators, singular values, trace class, trace.
L (22.4), 30.1, 30.2, 30.5 (partly), 30.8
- 29/9: Sobolev spaces, the Sobolev embedding theorem,
Sobolev's inequality. L p. 40 and 42 (and more).
- 30/9: The Riesz-Thorin interpolation theorem.
The Marcinkiewicz interpolation theorem.
L 16.2 (and more)
- 8/10:
PROBLEM SOLVING: Homework 2;
L Exercises Ch. 30: 1, 2, 3, 4, 5, 6, 11(d)(e)(h)(i)(j)
- 12/10: PROBLEM SOLVING:
Exams
23 October 2008
and
12 January 2009.
Tentamen
Kolla
tentamensschema
för ev. ändringar.
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
|
| gauge
| Minkowskifunktional
|
| 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).