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• 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)