Functional analysis

There are no prerequisites.

Banach spaces. Linear operators and functionals on Banach spaces.The open mapping theorem. The closed graph theorem. The principle of uniform boundedness. The second dual.The adjoint operator on a Banach space . Weak topologies. The Banach-Alaoglu theorem.The Krein-Milman theorem on extreme points.Banach algebras. Ideals and quotients. The spectrum of an element. Riesz functional calculus. The Gelfand transform.C-algebras. Approximate units. Ideals and quotients. Commutative C-algebras. The functional calculus in C*-algebras. The Gelfand-Naimark-Segal construction.

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.
M. Hladnik: Naloge in primeri iz funkcionalne analize in teorije mere, DMFA-založništvo, Ljubljana, 1985.
R. Meise, D. Vogt: Introduction to Functional Analysis, Oxford Univ. Press, Oxford, 1997.
G. K. Pedersen: Analysis Now, Springer, New York, 1996.
W. Rudin: Functional Analysis, 2nd edition, McGraw-Hill, New York, 1991.
I. Vidav: Linearni operatorji v Banachovih prostorih, DMFA-založništvo, Ljubljana, 1982.
• I. Vidav: Banachove algebre, DMFA-založništvo, Ljubljana, 1982.
I. Vidav: Uvod v teorijo C*-algeber, DMFA-založništvo, Ljubljana, 1982.

Students learn the basics of functional analysis and links with other areas of analysis.

Knowledge and understanding: Understanding
basic concepts of functional analysis. Ability of the reconstruction (at least easier) proofs. Ability of the application of acquired knowledge.
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. 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]
DRNOVŠEK, Roman. An infinite-dimensional generalization of Zenger's lemma. Journal of mathematical analysis and applications, ISSN 0022-247X. [Print ed.], 2012, vol. 388, iss. 2, str. 1233-1238. [COBISS-SI-ID 16214617]
Igor Klep:
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]