- Lecture 01. 07.01 Main definitions: Normed and metric spaces, Banach spaces, convergence,
linear operators and functionals, boundedness, dual spaces, examples. (Kalton and Bowers, Chapter 1)
- Lecture 02. 11.01 Spaces of sequences, examples, l^p, Young, Holder, and Minkovsky inequalities, norm
equivalence, unit balls in various l^p (Lecture notes)
- Lecture 03. 14.01 Dual for spaces of sequences (Lecture notes, Vershynin Section 2.2.3.)
Crash course of measure theory (beginning) (Bowers and Kalton Appendix A, lecture notes)
- Lecture 04. 18.01 Crash course of measure theory (cont). L^p spaces. (Bowers and Kalton Appendix A, lecture notes)
- Lecture 05. 21.01 Crash course of measure theory (end). L^p spaces: completeness, dual. Dual to C[0,1] (no proof yet)
(Bowers and Kalton Appendix A, lecture notes)
- Lecture 06. 25.01 Inner product spaces, Hilbert spaces. Cauchy-Schwartz, parallelogram relation, polar form. Orthogonality,
orthogonal projection, orthogonal decomposition. Riesz representation theorem. (Vershynin notes)
- Lecture 07. 31.01 Orthonormal systems, Bessel, Parseval relations. Complete orthogonal systems.
Separable Banach spaces. Gramm-Schmidt orthogonalization procedure. All separable Hilbert spaces are isometric.
Examples of orthonormal systems. (Vershynin notes)
- Lecture 08. 04.02 Partially ordered sets, upper, lower bounds etc. Zorn lemma. Formulation of the Hahn-Banach theorem. Proof. Supporting functionals, Banach limits, functionals separate points; completeness theorems. Sublinear functionals. Hahn-Banach theorem for sublinear functionals
- Lecture 09. 07.02 Second dual. Reflexive spaces. Application of Hahn-Banach theorem: the trigonometric system is dense in L^2(-\pi,\pi). Quotient spaces.
- Lecture 10. 11.02 Baire category theorem. Uniform boundedness principle.
- Lecture 11. 14.02 Banach-Steinhaus theorem. Weak boundedness. Conjugate operator. Examples.
- Lecture 12. 25.02 Open mapping theorem. Banach theorem on inverse operator. Closed graph theorem.
- Lecture 13. 28.02 Compact sets in metric spaces. Equivalence of three definitions. Properties of compact sets.
- Lecture 14. 04.03 Compact sets in various Banach spaces. Shauder basis. Approximation of compact sets by finite-dimensional ones.
- Lecture 15. 07.03 Arzela-Ascoli theorem. Weak convergence. Properties of weak convergent sequences. Criteria for weak convergence.
- Lecture 16. 08.03 Notion of topology. Examples. Topological vector spaces. Examples. Weak topology, countable norm topology. Example: tempered distributions. Weak* topology. Alauglu theorem (with no proof).
- Lecture 17. 11.03 Compact operators: definition, examples. Finite rank operators, approximation theorem. Diagonal operators in l^p. Integral operators. Green function for simplest differential equation.
- Lecture 18. 14.03 Properties: composition, sum. Algebraic digression: rings, ideals. Shauder theorem: compactness of the adjoint operator. Compact operator transforms weak convergence into strong convergence. Operator on quotient space. Image of compact operator cannot be closed, compact operator cannot be invertible.
- Lecture 19. 15.03 Fredholm theory: Formulation. Image of compact perturbation of the unite operator is closed. Fredholm alternative with zero kernel. The general case (exercise).
- Lecture 20. 18.03 Definition of spectrum. Examples. Classification of points of spectrum. Resolvent function. Neumann series, Spectrum is bounded. Perturbation of invertible operator. Spectrum is closed.
- Lecture 21. 21.03 Analytic operator functions. Reminder: the main theorem of algebra, Liouville theorem. Spectrum is not empty. Spectral radius. Spectrum of operator of integration.
- Lecture 22. 21.03 Spectrum of compact operators. Invariant subspaces. Hilbert-Schmidt operators.
- Lecture 23. 25.03 Integral operators as Hilbert-Schmidt operators. Unitary operators, their spectrum properties. Polar form (reminder). Selfadjoint operators, definition. Exponent of operator. Positive and negative operators. Polar form (with no proof)
- Lecture 24. 28.03 Selfadjoint operators: Point spectrum is real. No residual spectrum. Estimate from below. Continuous spectrum is real. Spectral interval. Operator norm is defined by special interval. Orthogonality of eigenvectors, corresponding different eigenvalues. Spectral theorem for compact selfadjoint operators. Hint about the general spectral theorem for selfadjoint operators.
- Lecture 25. 29.03 Functions of selfadjoint operators: polynomials of operators, arithmetic rules, spectral mapping theorem. Reminder: Weierstrass theorem, Muntz theorem (sketch of the proof). Continuous functions of operator. Convergence, spectral mapping theorem.
- Lecture 26. 01.04 Dunford calculus. Functions analytic in a vicinity of spectrum. Definition of function of operator. Arithmetic rules: addition, multiplication. Spectral mapping theorem. Composition of functions. Adjoint operator.
- Lecture 27. 04.04 Summary.