# TMA4230 Functional analysis, Spring 2015

Lecturer: Franz Luef.

# Exams

## May 22nd

, 10:00, room 940 (my office): Abdullah Abdulhaqu

## May 26th, room 922

## Schedule

- 09:00-09:45: Cristiano Gratton
- 09:45-10:30: Luis Perez
- 10:30-11:15: Haukern Holm Gulbrandsrud
- 11:15-12:00: Are Austad
- 12:00-12:45: Erik Skrettingland

## June 4th, room 922

## Schedule

- 09:00-09:45: Olav Willumsen Haugå
- 09:45-10:30: Petter K. Nyland
- 10:30-11:15: Egil A. Bugge
- 11:15-12:00: Håkon Bølviken
- 12:00-12:45: André Flakke
- 12:45-13:00: break
- 13:15-14:00: Audun Reigstad
- 14:00-14:45: Ola Mæhlen
- 14:45-15:30: Alexander N Sigurdsson
- 15:30-16:15: Yu Hengxiang

### Lectures

Mondays 12:15 - 14:00 in R60 (Realfagbygget, romnr: E3-103).

Tuesdays 10:15 - 12:00 in MA23 (Grønnbygget, romnr: 007).

Week: 2-13 & 16-17

Course will be given in English; course description

The course was supposed to follow the book by **Steven G. Krantz: A guide to functional analysis**, but in January a new book has appeared A. Bowers and N.J. Kalton **An Introductory Course in Functional Analysis**, which is much better suited for the course than Krantz's book. Hence I have taking the liberty to follow his text in the treatment of the Big Three Theorems and the Theorem of Hahn-Banach.

Some additional material might be taken from **Peter D. Lax: Functional analysis** which is highly recommended to anyone with an interest in functinonal analysis and its various applications. Peter D. Lax was the recipient of the Abel Prize in 2005. Another set of excellent notes on Functional analysis are by Roman Verschynin Lecture Notes -- Verschynin

At the beginning of the course there will be a brief review of the basic concepts of general topology, so that a proper treatment of some of the key notions of functional analysis is possible, e.g. weak and weak-* topology, and the Theorem of Banach-Alaoglu.

**Week 2**Basic facts on topological spaces: open sets, closed sets, neighbourhoods, subbasis and basis of a topology, product and quotient topology, closure and interior of a set. Compactness, open coverings, Tychonov's theorem. Metric spaces, total-boudedness.**Week 3**Complete metric spaces, Cauchy sequences, completion of a metric space, theorem of nested balls, Baire's category theorem, dense, nowhere dense, first categroy, second category. Normed and Banach spaces, sequence spaces, continuous functions, criterion for completeness of a normed space in terms of absolutely convergent series.**Week 4**Series and bases in normed spaces, Weierstrass approximation theorem. Linear operators, basic definitions.**Week 5**Uniform boundedness principle, and as application: existence of a nowhere differentiable continuous function.**Week 6**Open mapping theorem, bounded inverse mapping theorem, equivalent norms, Mazur's theorem on separable Banach spaces embedded as quotients into the space of absolutely convergent series.**Week 7**Dual spaces of sequence spaces. Zorn's lemma. Hahn-Banach for normed spaces.**Week 8**Hahn-Banach for vector spaces with sublinear functionals. Consequences of Hahn-Banach: Existence of supporting functional/norming element, natural embedding of a normed space into its bidual, a normed space is reflexive iff its dual is reflexive, separation of closed subspace and a point, adjoint operator for mappings between normed spaces, adjoint is an isometry, double adjoint of a bounded normed linear operator.**week 9**Duals of quotients and subspaces. Minimal approximants for closed, convex subsets of Hilbert spaces, Projection theorem, orthonormal series, abstract Fourier series, Bessel inequality and Parseval identity, completeness of orthonormal series, best approximation of Fourier series, Riesz representation theorem for linear functionals of a Hilbert space.**week 10**Existence of orthonormal basis for Hilbert spaces, isomorphisms between separable Hilbert spaces, Hilbert space adjoint, definition and basic properties of topological vector spaces. Dual of a topological space and characterization of bounded linear functionals. Weak convergence, Characterization of weak and strong topology on a Banach space and its equivalence to the metriziability of the weak topology.**week 11**Weak-* topology, weak-star continuous linear functionals, weak topology on the dual space. Various results on continuity of linear functionals and operators with respect to the different topologlies. Infinite dimensional Banach spaces have no norm-compact unit balls, Banach-Alaoglu, Goldstine's theorem, reflexivitiy is equivalent to weak compactness of the unit ball.**week 12**Compact operators, definitions and basic properties, precompact sets in normed spaces, characterization of precompact sets for Banach spaces with a Schauder basis (examples are sequence spaces), Arzela-Ascoli, multiplication operators on sequence spaces and their compactness, integral operators on the space of continuous functions and on Lebesgue spaces, Hille-Tamarkin and Hilbert-Schmidt operators, Schauder's theorem on the compactness of the adjoint of a compact operator.**week 13**Compact perturbations of the identity, Riesz's lemma, spectral theorem for compact operators on Banach spaces, spectral theorem for hermitian compact operators on Hilbert spaces.**week 14**Easter break**week 15**Easter break**week 16**Hilbert-Schmidt operators and their relation to their eigenvalues, Spectrum of bounded linear operators on normed spaces, resolvent set and different types of spectra (continuous, discrete and residual), Neumann series, resolvent identity, resolvent operator is analytic, spectrum is compact and non-empty, resolvent set is open, spectral radius formula.**week 17**Reviewing the main parts of the syllabus

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### Exercises

Thursdays 13:15 - 14:00 in R91 (Realfagbygget, romnr: DU1-145).

Week: 3-13 & 15-16 For a deeper understanding of the material it is essential to work out some problems. Hence it is strongly recommended to participate in the problem scessions on Thursdays. Please, contact me about any typos in the problem set or if some of the problems are not stated correctly. Note that you are not given any specific problems from the list below during your exam. There might be some problems discussed during the exam, but these are just have the character of examples.

Due to the courtesy of Cristiano Gratton and Petter Nyland there is a complete list of solutions to the problems.

- week 3: Problems 1-5.
- week 4: Problems 8,14,
- week 5: Problems 6,7,9
- week 6: Problems 10-15
- week 7: Problems 16-17
- week 8: Problems 18-20
- week 9: Problems 21-26
- week 10: Problems 26-29
- week 11: Problems 30-34
- week 12: Problems 35-38
- week 13: Problems 39-41
- week 14: Easter break
- week 15: Problems 41-42
- week 16: Problems 43-46

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### Reference group

- Are Austad
- Egil Aleksander Bugge

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## Contents of the course

This course is in many ways a continuation of the course TMA4145 Linear Methods, where the following material is covered Lecture Notes. The main subjects are Banach spaces, aka complete normed vector spaces and bounded linear operators on normed vector spaces. Highlights of the course include the following:

- Banach-Steinhaus theorem also known as the uniform boundedness principle.

## Literature

- Steven G. Krantz:
*A Guide to Functional Analysis*, ISBN 9781614442134. - Erwin Kreyszig:
*Introductory functional analysis with applications*, ISBN 0471504599. - Harald Hance-Olsen:
*Assorted notes on functional analysis*. - A brief English–Norwegian dictionary made by Harald Hance-Olsen covering some much used terms.