# TMA4230 Functional analysis, Spring 2014

## Teacher

- Lecturer: Franz Luef.

## Exam

- Here are some of the main topics and notions that you should be familiar with for the exam: topics.pdf

### Lectures

- Tuesdays 12:15 - 14:00 in VA2.
- Fridays 14:15 - 16:00 in R21.

Lectures are from Week 2 - **Week 15**

- week 2: Overview and some history of functional analysis. Definition and examples of metric spaces, normed spaces, innerproduct spaces, Hilbert spaces, Banach spaces. Notions of separability, dense subsets of a normed space. Sequence spaces and spaces of continuous functions.
- week 3: Baire category theorem and its consequences of pointwise limits of continuous functions. Notions: osicillation of a function, nowhere dense subsets, sets of 1st and 2nd category.
- week 4: Brief introduction to Lebesgue spaces, the inequalities of Hoelder, Minkowski and Young. The case of sequence spaces is covered in Sect. 1.2-1.3. Geometry of Hilbert spaces is from Chapter 3, Sections 3.1-3.6.
- week 5: Hilbert spaces: Section 3.5-3.6. Uniform Boundedness Principle also known as Banach-Steinhaus Theorem: Section 4.7, Open Mapping Theorem: 4.12. Closed operators: Section 4.13.
- week 6: Continuation of the discussion of Closed Graph Theorem, Open Mapping Theorem, Banach-Steinhaus Theorem, and some of its applications. (Section 4.7,4,12,4.13.) Discussion that Hilbert adjoints of an operator are closed operators.
- week 7: Hahn-Banach Theorem, Zorn's Lemma: Sections 4.1-4.3., adjoint operator, second dual and reflexive spaces: Sections 4.5-4.6.
- week 8: Further discussion of second dual: Section 4.6.
- week 9: Geometric version of Hahn-Banach, separation theorems, Minkowski functional, basic notions of convex geomtry. Strong and weak convergence in normed spaces (Section 4.8).
- week 10: Compactness in Banach spaces with a Schauder basis, strong and uniform convergence of operators (Section 4.9). Basic results on weak convergence, Schur property for the space of absolutely summable sequences.
- week 11: Weak-*-convergence, Banach-Alaoglu for seperable Banach spaces. Compact operators (Section 8.1)
- week 12: Compact operators (Section 8.1, 8.2.). Integral operators on the interval [0,1]: Boudedness conditions for various Lebesgue spaces and compactness for square-integrable functions (Hilbert-Schmidt condition). Adjoint of an integral operator.
- week 13: Adjoint of an operator in the case of normed spaces. Schauder's theorem (Section 8.3). Compact operators are the norm completion of the finite rank operator in the case of Hilbert spaces. Noninvertibility of compact operators, Eigenspaces of eigenvalues of compact operators are finite-dimensional.
- week 14: Banach algebra (Section 7.6,7.7) and spectral theory (Section 7.1-7.5)
- week 15: Spectral theory of compact operators

### Problem session

- Mondays 15:15 - 16:00 in F3.

- Problem set 1: problemset_1.pdf
- Problem set 2: Section 3.3: Problems 6 & 7, Section 3.4: Problems 5-10, Section 3.6: Problem 4.
- Problem set 3: Section 4.7: Problems 11, 12, 13, 14, 15.
- Problem set 4: Section 3.9: Problems 3, 8, 10. Section 4.2: Problems 5, 6, 10. Section 4.3: Problems 2, 4, 8. Section 4.6: Problems 4, 9.
- Problem set 5: Section 4.8: Problems 2,3,4,9. Section 4.9: Problems 1,2,3,4,7.
- Prolbem set 6: Section 7.6 Problems 1,2,3,4,6,10.

## Language

This course will be given in English.

## Links

- The official course description.
- A brief English–Norwegian dictionary made by Harald Hance-Olsen covering some much used terms.

## Contents of the course

This course is in many ways a continuation of the course TMA4145 Linear Methods. The main subjects are complete normed vector spaces and bounded (continuous) linear operators on normed vector spaces. Highlights of the course include the following:

## Literature

- Erwin Kreyszig:
*Introductory functional analysis with applications*, ISBN 0471504599. - Harald Hance-Olsen:
*Assorted notes on functional analysis*.