Mathematics of quantum mechanics - methods of Hilbert spaces

Topisc covered

  1. Banach spaces of continuous functions
  2. Construction of the spaces \(L^p(\Omega,\mu)\), proof of completeness
  3. Finite-dimensional spaces
  4. Sesquilinear forma, polarization identity, Schwarz inequality, scalar product and norm
  5. Orthonormal systems, convergence of series with square summable coefficients, Bessel inequality, orthonormal bases
  6. Approximation by continuous functions (applications of theorems of Lusin and Stone-Weierstrass), examples of orthonormal bases
  7. Orthogonal complements, the orthogonal projection theorem
  8. The operator \(P_\mathcal{M}\), examples

Lecture notes (in Polish)