Skip to main content

Introduction to functional analysis

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M1
ECTS:
6
Language:
slovenian, english
Lecturer (contact person):
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Prerequisites

There are no prerequisites.

Content (Syllabus outline)

Hilbert spaces. Orthonormal systems. Bessel's inequality. Completeness. Fourier series. Parseval's identity.
Linear operators and functionals on Hilbert spaces.The representation of a continuous linear functional.Adjoint operator. Selfadjoint and normal operators.Projectors and idempotents. Invariant subspaces.Compact operators. The spectrum of a compact operator.Diagonalization of a selfadjoint compact operator.An application: Sturm-Liouville systems.Banach spaces. Examples.Linear operators and functionals on Banach spaces.Finite dimensional normed spaces. Quotients and products of normed spaces.The Hahn-Banach theorem and consequences. Separation of convex sets.

Readings

B. Bollobás: Linear Analysis : An Introductory Course, 2nd edition, Cambridge Univ. Press, Cambridge, 1999.
J. B. Conway: A Course in Functional Analysis, 2nd edition, Springer, New York, 1990.
Y. Eidelman, V. Milman, A. Tsolomitis: Functional Analysis : An Introduction, AMS, Providence, 2004.
D. H. Griffel: Applied Functional Analysis, Dover Publications, Mineola, 2002.
M. Hladnik: Naloge in primeri iz funkcionalne analize in teorije mere, DMFA-založništvo, Ljubljana, 1985.
E. Zeidler: Applied Functional Analysis : Main Principles and Their Applications, Springer, New York, 1995.

Objectives and competences

Students acquire basic knowledge of the theory of Hilbert spaces and linear operators between them. The theory is applied for solving simple Sturm-Liouville problems. Students also learn some basic concepts from the theory of Banach spaces, which are a generalization of Hilbert spaces.

Intended learning outcomes

Knowledge and understanding: Understanding of the theory of Hilbert spaces.
Application: Functional analysis is used in natural sciences and other areas of science such as economics.
Reflection: Understanding of the theory on the basis of examples.
Transferable skills: Ability to use abstract methods to solve problems. Ability to use a wide range of references and critical thinking.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

Homeworks
Written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Roman Drnovšek:
DRNOVŠEK, Roman. An irreducible semigroup of idempotents. Studia Mathematica, ISSN 0039-3223, 1997, let. 125, št. 1, str. 97-99. [COBISS-SI-ID 7436633]
DRNOVŠEK, Roman. Common invariant subspaces for collections of operators. Integral equations and operator theory, ISSN 0378-620X, 2001, vol. 39, no. 3, str. 253-266. [COBISS-SI-ID 10597721]
DRNOVŠEK, Roman. Invariant subspaces for operator semigroups with commutators of rank at most one. Journal of functional analysis, ISSN 0022-1236, 2009, vol. 256, iss. 12, str. 4187-4196. [COBISS-SI-ID 15167321]
Peter Šemrl:
ŠEMRL, Peter, VÄISÄLÄ, Jussi. Nonsurjective nearisometries of Banach spaces. Journal of functional analysis, ISSN 0022-1236, 2003, vol. 198, no. 1, str. 268-278. [COBISS-SI-ID 12371545]
ŠEMRL, Peter. Generalized symmetry transformations on quaternionic indefinite inner product spaces: an extension of quaternionic version of Wigner's theorem. Communications in Mathematical Physics, ISSN 0010-3616, 2003, vol. 242, no. 3, str. 579-584. [COBISS-SI-ID 12770649]
ŠEMRL, Peter. Applying projective geometry to transformations on rank one idempotents. Journal of functional analysis, ISSN 0022-1236, 2004, vol. 210, no. , str. 248-257. [COBISS-SI-ID 13012825]
Igor Klep:
KLEP, Igor, VINNIKOV, Victor, VOLČIČ, Jurij. Local theory of free noncommutative functions: germs, meromorphic functions, and Hermite interpolation. Transactions of the American Mathematical Society. Aug. 2020, vol. 373, no. 8, str. 5587-5625. ISSN 0002-9947. [COBISS-SI-ID 23631107]
HELTON, J. William, KLEP, Igor, MCCULLOUGH, Scott. The tracial Hahn-Banach theorem, polar duals, matrix convex sets, and projections of free spectrahedra. Journal of the European Mathematical Society. 2017, vol. 19, iss. 6, str. 1845-1897. ISSN 1435-9855. [COBISS-SI-ID 18057817]
HELTON, J. William, KLEP, Igor, MCCULLOUGH, Scott, VOLČIČ, Jurij. Bianalytic free maps between spectrahedra and spectraballs. Journal of functional analysis. Jun. 2020, vol. 278, iss. 11, art. 108472 (61 str.). ISSN 0022-1236. [COBISS-SI-ID 15911683]