Introduction to functional analysis

There are no prerequisites.

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.

1. B. Bollobás: Linear analysis : an introductory course, 2nd ed., Cambridge : Cambridge Univ., cop. 1999.
2. J. B. Conway: A course in functional analysis, 2nd ed., New York : Springer, cop. 1990.
3. Y. Eidelman, V. Milman, A. Tsolomitis: Functional analysis : an introduction, Providence : American Mathematical Society, cop. 2004.
4. D. H. Griffel: Applied functional analysis, Chichester : Ellis Horwood ; New York : J. Wiley & Sons, cop. 1981.
5. M. Hladnik: Naloge in primeri iz funkcionalne analize in teorije mere, Ljubljana : Društvo matematikov, fizikov in astronomov SRS, 1985.
6. E. Zeidler: Applied functional analysis : main principles and their applications, New York : Springer, cop. 1995.

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.

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.

Lectures, exercises, homeworks, consultations

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

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]

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]