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Identical coursesAims
Functional analysis is the branch of mathematics dealing with vector spaces equipped with certain topologies and linear maps between them. It is a very important part of modern analysis. This course is a master level introduction to this area of mathematics.
Historically, the field of functional analysis arose from the study of spaces of functions, which still serve as motivating examples. The course includes an introduction to spectral theory for Hilbert space operators. The abstract results on topological vector spaces, Banach spaces and Hilbert spaces will be illustrated with examples and applications coming from different areas of mathematics. Apart from being an important area of theoretical mathematics, functional analysis provides mathematical background for e.g. theoretical physics, partial differential equations and optimization, but these topics will not be covered in the course.
After following this course, the student
- is able to independently give proofs of results related to the course material,
- is able to apply the course material in different areas of mathematics,
- is able to learn himself/herself a new concept in functional analysis,
- is able to study advanced texts in functional analysis.
Previous knowledge
The student should be familiar with advanced and rigorous analysis (as covered, for example, in Analyse II
(B-KUL-G0N86B)), including Lebesgue integration for functions of one and several variables and the notion of Hilbert space. Prior knowledge on abstract measure theory is not necessary. Students should be familiar with general topology. The course Topologie
(B-KUL-G0P55B) definitely suffices.
Is included in these courses of study
Activities
5 ects. Functional Analysis (B-KUL-G0B03a)
Content
Hilbert spaces and Banach spaces
- Reminders on Hilbert spaces, orthogonal projections and orthonormal bases
- Definitions, examples and basic properties of Banach spaces
Baire category and its consequences
- Baire category theorem
- Boundedness and continuity of linear maps
- Open mapping theorem
- Closed graph theorem
- Principle of uniform boundedness
Bounded operators on a Hilbert space
- Hermitian adjoint
- Compact operators
- Invertible operators, spectrum of an operator
- Spectral theory of compact selfadjoint operators
- Spectral theory of arbitrary selfadjoint operators
Weak topologies and locally convex vector spaces
- Dual Banach space
- Hahn-Banach extension theorem
- Topological vector spaces, seminormed spaces
- Weak topologies
- Hahn-Banach separation theorem
- Banach-Alaoglu theorem
- Krein-Milman theorem
- Markov-Kakutani fixed point theorem
Amenability of groups
- Invariant means on groups
- Examples and counterexamples to amenability
- Various characterizations of amenability
- Abelian groups are amenable
Course material
Lecture notes.
Book: G.K. Pedersen, Analysis Now. Graduate Texts in Mathematics, Volume 118. Corrected second printing. Springer-Verlag, New York, 1995.
Format: more information
There will be a two-hour lecture each week, in which new concepts will be introduced and several results will be proved. Additionally, there will be an exercise session each week, in which the students further develop the topics of the course and apply the material in different situations.
1 ects. Functional Analysis: Exercises (B-KUL-G0B04a)
Content
Exercises and problem assignments related to the different topics of the course.
Course material
Lecture notes.
Book: G.K. Pedersen, Analysis Now. Graduate Texts in Mathematics, Volume 118. Corrected second printing. Springer-Verlag, New York, 1995.
Evaluation
Evaluation: Functional Analysis (B-KUL-G2B03a)
Explanation
Detailed information will be provided via Toledo.
Information about retaking exams
For the second exam opportunity the grades on the take-home assignments are transferred. There is no possibility to have a new take-home assignment.