Methods of Hilbert Spaces – lecture material

  1. Normed space, Banach spaces, examples
  2. \(\mathsf{L}_p\) spaces, proof of completeness
  3. Linear operators, norm of na operator, boundedness, the space \(\operatorname{B}(\mathsf{X},\mathsf{Y})\)
  4. Sesquilinear forms, scalar products, polarization formula
  5. Schwarz inequality, the norm, parallelogram rule
  6. Gram-Schmidt orthogonalization
  7. Series of vectors in normed spaces
  8. Convergence of series of orthonormal vectors with square-summable coefficients
  9. Bessel's inequality
  10. Equivalent characterizations of complete orthonormal systems
  11. Examples of orthonormal bases
  12. Approxiamtion by continuous functions in \(\mathsf{L}_p\) spaces, theorems of Lusin and Stone-Weierstrass
  13. Examples of orthonormal bases
  14. The orthogonal complement and orthogonal projections
  15. Vector minimizing distance to a closed convex subset of a Hilbert space
  16. Riesz representation theorem and applications
  17. Sesquilinear forms and operators
  18. The adjoint operator, examples: projections, ket and bra
  19. Various equivalent characterizations of isometric and unitary operators
  20. Finite and infinite direct sums of Hilbert spaces
  21. Bijection between closed subspaces of \(\mathcal{H}\) and self-adjoint idempotents in \(\mathrm{B}(\mathcal{H})\)
  22. Tensor products of vector spaces and of Hilbert spaces
  23. Examples of tensor products, symmetric and antisymmetric tensor products, Fock spaces
  24. Fourier series of square-integrable functions
  25. Fourier series of integrable and integrable functions, Riemann-Lebesgue lemma
  26. Injectivity of the map \(L_1(\mathbb{T})\ni{f}\mapsto\widehat{f}\in\mathrm{c}_0(\mathbb{Z})\)
  27. Convolutions on \(\mathbb{T}\), multiplicativity of the map \(\mathsf{L}_1(\mathbb{T})\ni{f}\mapsto\widehat{f}\in\mathrm{c}_0(\mathbb{Z})\)
  28. Existence of continuous functions with Fourier series divergent at a point
  29. Fourier series of continuous functions, Fejér's Theorem
  30. Other convergence results: Fourier series of \(\mathrm{C}^1\) functions, Dini's theorem, Fourier series of Hölder continuous functions
  31. The Fourier transform of integrable functions on \(\mathbb{R}^d\)
  32. The Schwartz space, embeddings into \(\mathsf{L}_p(\mathbb{R}^d)\)
  33. Riemann-Lebesgue lemma
  34. Fourier inversion theorem
  35. Application of injectivity of the ourier transform to orthogonal polynomials
  36. The Fourier transform on \(\mathsf{L}_2(\mathbb{R}^d)\)
  37. Definition of the Sobolev spaces \(\mathsf{H}^s\), Sobolev embedding theorem for \(\mathsf{H}^s\)
  38. Radon-Nikodym theorem

Lecture notes

  1. Lecture 1 (in Polish)
  2. Lecture 2 (in Polish)
  3. Lecture 3 (in Polish)
  4. Lecture 4 (in Polish)
  5. Lecture 5 (in Polish)
  6. Lecture 6 (in Polish)
  7. Lecture 7 (in Polish)
  8. Lecture 8 (in Polish)
  9. Lecture 9 (in Polish)
  10. Lecture 10 (in Polish)
  11. Lecture 11 (in Polish)
  12. Lecture 12 (in Polish)
  13. Lecture 13 (in Polish)
  14. Lecture 14 (in Polish)
  15. Lecture 15 (in Polish)

Recommended literature

  1. M. Reed, B. Simon – Methods of modern mathematical physics Vol. I
  2. K. Maurin – Methods of Hilbert spaces
  3. G.K. Pedersen – Analysis now