# Functional Analysis / Funktionaalianalyysin peruskurssi, Spring / Kevät 2017

**Teacher / Vastuuopettaja:** Jari Taskinen

## News / current

The grading of the course has been completed and the results have been registered to web-oodi.

## Information on the course

The lectures are on the weeks 3-9 and 11-18, Mondays 14-16 and Thursdays 14-16 in room C124. Easter holiday is on April 13th-19th. The course corresponds to 10 credit points.

The course is on the advanced studies level. Prerequisites include differential and integral calculus in one and several variables, linear algebra, metric space theory, and preferably also introduction to Lebesgue measure and integration theory.

Functional analysis means analysis in infinite dimensional spaces. Interesting objects include Banach- and Hilbert-spaces and linear operators between such spaces. Most important examples of Banach spaces are various sequence and function spaces, for example L^p- and Sobolev-spaces. Fourier-transform, Laplace-operator and shift operator are examples of linear operators.

We shall consider basic properties and most important examples of Banach- and Hilbert-spaces and their linear operators. We prove the three basic principles of linear functional analysis and consider applications for example to differential equations. This material is important for example in

- Complex and harmonic analysis, where one uses Hardy- and Bergman-spaces and Fourier- and Hilbert-transforms,
- Applied mathematics, where for example the theory of elliptic partial differential equations is nowadays written in the language of Hilbert-space linear operators,
- Mathematical physics, with numerous applications, for example in the spectral theory of the Schrödinger equation.

## Exams

There will be two course exams. The first exam is on Thu. March 2nd at 14.15-16.00 in the class C124. The second course exam is on Tuesday May 2nd at 14.15-16.00 in B119. The material for the second examination includes the lectures from the week 9 onwards (starting from Chapter 5, Fejér kernel, of the lecture notes) and Exercises 6-12. Answers are accepted in Finnish, Swedish or English. Concerning calculators, tables and other devices, standard rules of the department apply.

The maximum of each exam is 24 points, and to pass the course one has to get the minimum of 8 points in each exam (in addition to the usual minimum of the total point number). Bonus points from solutions of exercises: 25 % of problems solved = 1 point, 35 % = 2 points, 45 % = 3 points, 55 % = 4 points, 65 % = 5 points, 75 % = 6 points, to be added to the results of examinations.

## Course material

Funktionaalianalyysin peruskurssi, luentomoniste

Friedman, A., Foundations of Modern Analysis, Dover 1982.

Conway, J. A Course in Functional Analysis. Springer, 1990. (Introduction to the topic)

Maddox, I.J., Elements of Functional Analysis, Cambridge University Press, 1977.

Rudin, W., Functional Analysis. McGraw Hill 1974. (Quite difficult and general)

Brezis, H., Analyse fonctionnelle, Masson, Paris 1993. (In French. Lot of PDE applications)

Werner, D., Funktionalanalysis, Springer Lehrbuch 1990. (In German)

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## Exercise problems as pdf-files

### Execrcise groups:

Group | Day | Time | Room | Teacher |
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1. | Fri. | 14-16 | C124 | Roberta Bosi |

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