Introduction to functional analysis

Enrollment into the program.

Positive result from qoloquia (or written exam) is necessary to enter the oral exam.

Hilbert space. Orthonormal systems. Fourier series.
Bessel's inequality. Completeness. Parseval's theorem.
Linear operators and functionals on Hilbert spaces.
Representation theorem for linear functionals.
Adjoint operator. Selfadjoint and normal operators.
Projectors and idempotents. Invariant subspaces.
Compact operators. Spectrum of compact operators.
Ries'z decomposition of compact operators.
Diagonalizability of compact selfadjoint operators.
Application: Sturm-Liouville problem.

J. B. Conway: A Course in Functional Analysis, 2nd edition, Springer, New York, 1990.
D. H. Griffel: Applied Functional Analysis, Dover Publications, Mineola, 2002.
E. Zeidler: Applied Functional Analysis : Main Principles and Their Applications, Springer, New York, 1995.

Objectives: Student gets familiar with the basic facts about Hilbert spaces and operators acting on Hilbert spaces. Sturm-Liouville problem is presented in order to see possible applications.
Competences: Knowledge and understanding of basic principles of functional analysis.

Knowledge and understanding:
Understanding of Hilbert space theory from both theoretical and applied aspects.
Application:
Functional Analysis can be applied in Sciences, Technology, and even in Social Sciences.
Reflection:
Understanding of mathematical theory based on applications.
Transferable skills:
Identification of problems and solving them. The formulation of nonmathematical
problems in the language of mathematics. The use of adequate literature in Slovene and foreign languages.

Lectures, exercises, homeworks, consultations

2 midterm exams or final written exam
grading homeworks, oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

KLEP, Igor, Štrekelj, Tea. Facial structure of matrix convex sets. Journal of functional analysis. - ISSN 0022-1236 (Vol. 283, iss. 7, Oct. 2022, art. 109601 (55 str.)) [COBISS-SI-ID 136207875]

KLEP, Igor, Vinnikov, Victor, Volčič, Jurij. Local theory of free noncommutative functions: germs, meromorphic functions and Hermite interpolation. Transactions of the American Mathematical Society. - ISSN 0002-9947 (Vol. 373, no. 8, Aug. 2020, str. 5587-5625) [COBISS-SI-ID 23631107]

Helton, J. William, KLEP, Igor, McCullough, Scott, Schweighofer Markus. Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions. znanstvena monografija, Providence : American Mathematical Society, cop. 2019 ISBN - 978-1-4704-3455-7; 978-1-4704-4947-6 [COBISS-SI-ID 18571865]