Функциональный анализ и теория операторов

Lecture 1. Basics of Functional Analysis. Metric Spaces

Lecture 2. Metric Spaces. Normed Spaces. Seminorms and Polynormed Spaces. Banach Spaces

Lecture 3. Euclidean and Hilbert Spaces

Lecture 4. Separable Hilbert Spaces. Bases in Hilbert Spaces

Lecture 5. Compact and Precompact Sets in Metric Spaces

Lecture 6. Compact and Precompact Sets: Exercises

Lecture 7. Linear Operators and Functionals in Normed Spaces

Lecture 8. Linear Operators and Functionals in Normed Spaces: Exercises

Lecture 9. The Hahn–Banach Theorem and the Corollaries

Lecture 10. (C[a,b])*. Norms of Functionals

Lecture 11. Hilbert Space Duality. Modes of Convergence

Lecture 12. Reproducing Kernels and Weak Convergence: Exercises

Lecture 13. Adjoint, Self-Adjoint, and Normal Operators. Hellinger–Toeplitz Theorem

Lecture 14. Adjoint Operators: Exercises

Lecture 15. Compact Operators. Inverse Operator

Lecture 16. Exercises on Compact and Inverse Operators

Lecture 17. Spectrum of a Bounded Operator. Classification of Points in the Spectrum

Lecture 18. Exercises on Spectra of Operators

Lecture 19. The Hilbert–Schmidt Theorem

Lecture 20. Applications of the Hilbert–Schmidt Theorem

Lecture 21. Fredholm Theory

Lecture 22. Fredholm Theory: Exercises

Lecture 23. Unbounded Operators: Introduction

Lecture 24. Symmetric Operators

Lecture 25. Isometric Operators and the Cayley Transform. Self-Adjoint Extensions of Symmetric Operators

Lecture 26. Functional Calculus

Lecture 27. Spectral Theorem for Self-Adjoint Operators. Fourier Transform in L₁

Lecture 28. Fourier Transform in L₁, S, and L₂

Lecture 29. Test Functions and Distributions

Lecture 30. Convolution