Functional analysis

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

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.

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.
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.

Objectives and competences

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

Intended learning outcomes

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.

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. 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]
Peter Šemrl:
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]
ŠEMRL, Peter. Similarity preserving linear maps. Journal of operator theory, ISSN 0379-4024, 2008, vol. 60, no. 1, str. 71-83. [COBISS-SI-ID 15079257]
ŠEMRL, Peter. Symmetries on bounded observables: a unified approach based on adjacency preserving maps. Integral equations and operator theory, ISSN 0378-620X, 2012, vol. 72, iss. 1, str. 7-66. [COBISS-SI-ID 16568665]