Introduction to functional analysis

There are no prerequisites.

Banach spaces. Examples. Finite dimensional normed spaces. Quotients and products of normed spaces. Linear operators and functionals on Banach spaces. Dual space. The Hahn-Banach theorem and consequences. Separation of convex sets. The open mapping theorem. The closed graph theorem. Principle of uniform boundedness. 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.

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. 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 Bessonov:

• BESSONOV, Roman V., Szegő condition and scattering for one-dimensional Dirac operators, Constructive approximation. - ISSN 0176-4276 (Vol. 51, iss. 2, Apr. 2020, str. 273-302), COBISS-SI-ID - 218704899
• BESSONOV, Roman V., Wiener-Hopf operators admit triangular factorization. Journal of operator theory. - ISSN 0379-4024 (Vol. 82, iss. 1, 2019, str. 237-249), COBISS-SI-ID - 218617603
• BESSONOV, Roman V.; DENISOV, Sergey A., A spectral Szegő theorem on the real line, Advances in mathematics. - ISSN 0001-8708 (Vol. 359, [article no.] 106851, Jan. 2020, 41 str.), COBISS-SI-ID - 218696707

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]

Marko Kandić:

• KANDIĆ, Marko. On algebras of polynomially compact operators. Linear and Multilinear Algebra, ISSN 0308-1087, 2016, vol. 64, no. 6, str. 1185-1196. [COBISS-SI-ID 17493337]
• KANDIĆ, Marko. Ideal-triangularizability of nil-algebras generated by positive operators. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2011, vol. 139, no. 2, str. 485-490. [COBISS-SI-ID 15710809]
• DRNOVŠEK, Roman, KANDIĆ, Marko. Positive operators as commutators of positive operators. Studia Mathematica, ISSN 0039-3223, 2019, tom 245, str. 185-200. [COBISS-SI-ID 18407769]

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]

Aleksey Kostenko:

• KOSTENKO, Aleksey. Hankel Operators and Applications (Lecture Notes: https://users.fmf.uni-lj.si/kostenko/teach/HankelNotes.pdf), 2020; COBISS-SI-ID - 30308867.
• KOSTENKO, Aleksey, Trace Ideals and Applications (Lecture Notes: https://users.fmf.uni-lj.si/kostenko/teach/IdealsNotes.pdf), 2020; COBISS-SI-ID - 30316803.
• KOSTENKO, Aleksey, NICOLUSSI, Noema, Laplacians on Infinite Graphs, Memoirs of the European Mathematical Society, ISBN - 978-3-98547-025-9, 2023, vol. 3, 2023; viii+232 str.