Funktionalanalys I (5hp), VT 2012

Kurskod 1MA043

Lärare: Svante Janson

Kursplan        Studentportalen

Schema        Denna veckas schema

Kurslitteratur

Course contents

Conway:
Ch. I: 1 - 5.5
Ch. II: 1 (to 1.4), 2, 4 (to 4.9), 5 (only 5.4, 5.8, 5.9)
Ch. III: 1-3, 5-6, 12, 14.
Ch. IV: Theorems 3.3 and 3.9.
Ch. V: Definition 1.1.

Homework

Three homework assignments will be given. These form part of the examination, and should be solved individually. Correct solutions will give bonus points on the exam (up to 2 bonus points per set, i.e., total maximum 6). Solutions can be given to me at lectures, or in my mailbox at Ångström (floor 4), at the latest on the days shown below.
  1. 27 February.
  2. 5 March.
  3. 12 March.

Lectures

Here is a rough list of contents for the lectures, with approximate indications of sections in the textbook (Conway). Recommended exercises from the textbook are often given. They will be solved at a later lecture (marked PROBLEM SOLVING).
  1. 16/1: I.1. inner product, inner product spaces. Exerc. I.1.1.
  2. 18/1: I.2. Metric space, completeness, Hilbert space.
  3. 20/1: I.2. Orthogonality; orthogonal projection. Exerc. I.2:1, 2, 4, 5, 6
  4. 30/1: I.3. linear functionals in a Hilbert space; PROBLEM SOLVING
  5. 2/2: I.4. ON sets and bases. Exerc. I.4.1-3.
  6. 13/2: I.5 (to I.5.5), II.1 (to 1.4) II.2: Linear operators in a Hilbert space, adjoint, self-adjoint, normal, unitary, isometry, isomorphism. Exerc. I.5: 2-3,9; II.1: 1-4,8.
  7. 15/2: II.2: adjoints of linear operators. Exerc. II.2: 1-2
  8. 17/2: II.4 (to II.4.9) II.5 (only 5.4, 5.8, 5.9). Compact sets in metric spaces, compact operators, the spectral theorem for compact self-adjoint operators in a Hilbert space. Exerc. II.4: 1,2,5,6,8; II.5: 3,5
  9. 27/2: PROBLEM SOLVING
  10. 28/2: III.1-3. Banach spaces, linear operators. Exerc. III.1:1,2,5,16
  11. 1/3: III.5. Linear functionals. Dual space. Exerc. III.5.1-2.
  12. 5/3: III.6 (IV.3: Theorems 3.3, 3.9) The Hahn-Banach Theorem. Exerc III.6.6.
  13. 7/3: V.1, III.12. Weak and weak* convergence. The Open Mapping Theorem. The Closed Graph Theorem. Exerc III.12:4-5.
  14. 9/3: III.14. The Principle of Uniform Boundedness. The Banach-Steinhaus Theorem. Exerc III.14:2,6,9.
  15. 12/3: PROBLEM SOLVING Exam 2008-03-13.

Tentamen

Kolla tentamensschema för ev. ändringar.

Old exams (the course contents may have varied slightly)

Note the the course book was allowed on these exams, but it is not allowed this year.

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
contraction kontraktion
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
unbounded obegränsad
vector space vektorrum
På svenska skrivs Banachrum, Hilbertrum och andra sammansättningar med personnamn som ett ord.