# MAT-62256 Advanced Functional Analysis, 7 cr

#### Lisätiedot

Suitable for postgraduate studies. Ei toteuteta lukuvuonna 2018-2019.

#### Vastuuhenkilö

Lassi Paunonen, Esko Turunen

#### Opetus

 Toteutuskerta Periodi Vastuuhenkilö Suoritusvaatimukset MAT-62256 2018-01 - Lassi Paunonen Esko Turunen Final exam

#### Osaamistavoitteet

After completion of the course the student is mastering the foundations of the functional analysis the main results and is capable of deducing them from the definitions. The student can compute the generalized derivatives and apply them for solving partial differential equations. The student can explain different type of Sobolev spaces and knows their importance.

#### Sisältö

 Sisältö Ydinsisältö Täydentävä tietämys Erityistietämys 1. Topological foundations Banach- ja Hilbert spaces Hahn-Banach theorem The open mapping theorem Baire category theorems Closed operator theorems Weak topology. 2. Generalized measure theory Rieszin representation theorem The proofs involving results connecting Lebesgue and the generalized measure theory The proof of the Riesz represention theorem 3. Distribution theory and its applications to partial differential equations The exact definition of tthe topology of the space of distributions 4. Sobolev spaces

#### Ohjeita opiskelijalle osaamisen tasojen saavuttamiseksi

The grade of the course is based on the final exam or two partial exams. When the points for the final exam or the partial exams are 30% of the maximum, the grade of the course may be improved by bonus points collected from the instructed exercises and homework. The passing limit is 50% of the maximum. If the student is mastering the concepts, results, short proofs and examples type of problems the evaluation is 3. For the grade 4 the student should in addition to the previous level be able to independently apply theory more. For the grade 5 the student should independently deduce results, invent solutions and compare results more than in the previous levels.

#### Arvosteluasteikko:

Numerical evaluation scale (0-5)

#### Osasuoritukset:

Completion parts must belong to the same implementation

#### Oppimateriaali

 Tyyppi Nimi Tekijä ISBN URL Lisätiedot Tenttimateriaali Book Functional Analysis G. Bachman, L. Narici No Book Functional Analysis Lax No Book Functional Analysis Rudin No Summary of lectures MAT-53550 Advanced Functional Analysis Eriksson Yes

#### Esitietovaatimukset

 Opintojakso P/S Selite MAT-60206 Mathematical Analysis Advisable MAT-61006 Introduction to Functional Analysis Advisable MAT-61757 Measure and Integration Advisable

#### Vastaavuudet

 Opintojakso Vastaa opintojaksoa Selite MAT-62256 Advanced Functional Analysis, 7 cr MAT-53557 Advanced Functional Analysis, 8 cr

 Päivittäjä: Laaksonen Merja, 15.12.2018