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Functional analysis, spring 2016
The grading of the course will appear soon / has appeared in the weboodi; participants may inquire about details from Jari by e-mail.
Information on the course
The lectures are on the weeks 4-9 and 11-18, Mondays 14-16 and Thursdays 14-16 in room C124. Easter holiday is on 24.-30.3. 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.
There will be two exams: the first exam on Fri. March 11 at 12.15-14.45 in the auditorium A111, and the second on Fri. May 13th at 12.15-14.45 in A111.
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. 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.
Funktionaalianalyysin peruskurssi, luentomoniste
Rynne, B., Youngson, M., Linear Functional Analysis, Springer Undergraduate Mathematics Series, London, 2000. (Introduction to the topic)
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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