Mathematics BSc
Course description
2013.

Functional Analysis
Hours
lect+pc
Credits
lect+pc
Assessment Specialization Course code
lect/pc
Semester Status
2 + 2 2 + 3 exam +
term grade
pure math. mm1c1fa5m
mm1c2fa5m
5 compulsory
Strong Weak Prerequisites
Practice class
Strong:
Analysis2L (mm1c1an2) or
Foundations of analysisL (mm1c1ap2)
Lecture
Weak:
practice class
Weak:
Analysis3L-p (mm1c1an3m)
Prerequisites
Analysis 3, notion of Lebesgue integral, linear algebra.
Literature
  • F. Riesz, B. Szőkefalvi: Functional Analysis. Dover, 1990.
Syllabus
  • Hilbert spaces: inner product, Cauchy-Bunyakovsky-Schwarz inequality, 2, L2 spaces, distance from closed subspace, orthonormal basis, Fourier series, Bessel's inequality, Parseval's identity, continuous linear functionals on Hilbert spaces, Riesz's theorem.
  • Continuous linear operators in Hilbert spaces: norm, numerical radius, adjoint, spectrum, selfadjoint, normal, unitary operators, orthogonal projections, compact operators in Hilbert spaces, Hilbert-Schmidt theorem.
  • Banach spaces: distance from subspace, Riesz's lemma, Baire category theorem, continuous linear functionals, Hahn-Banach theorem, Mazur-Orlicz theorem, continuous linear operators, adjoint, spectrum, Neumann series, Banach-Steinhaus theorem, uniform boundedness principle, Banach open mapping theorem, closed graph theorem, compact operators in Banach spaces, Riesz-Fredholm theory, Schauder's theorem, Lomonosov's theorem.