Section: Course record | Functional Analysis A.A. 2024-2025

Section outline

- Select activity Nov 5: measure spaces, Borel sigma algebra, characterization of Borel measures on R
  
  Nov 5: measure spaces, Borel sigma algebra, characterization of Borel measures on R File
- Select activity Nov 12: measurable functions, pushforward of a measure, random variables, singular and absolutely continuous measures wrt Lebesgue, Lebesgue-Radon-Nikodym theorem
  
  Nov 12: measurable functions, pushforward of a measure, random variables, singular and absolutely continuous measures wrt Lebesgue, Lebesgue-Radon-Nikodym theorem File
- Select activity Nov 19: integration of functions of random variables. Jensen inequality. Normed space, and distance associated to a norm. Complete normed space. Examples
  
  Nov 19: integration of functions of random variables. Jensen inequality. Normed space, and distance associated to a norm. Complete normed space. Examples File
- Select activity November 26: Banach spaces. Examples: L^p spaces. Holder inequality. Definition of scalar product and Hilbert spaces
  
  November 26: Banach spaces. Examples: L^p spaces. Holder inequality. Definition of scalar product and Hilbert spaces File
- Select activity December 3: space of random variable with finite variance. Conditional expectation as the orthogonal projection. Best estimator of a random variable
  
  December 3: space of random variable with finite variance. Conditional expectation as the orthogonal projection. Best estimator of a random variable File
- Select activity December 10: orthogonal projection in finite dimensional subspaces. The product of convolution. Fourier transform and main properties
  
  December 10: orthogonal projection in finite dimensional subspaces. The product of convolution. Fourier transform and main properties File
- Select activity December 17: Fourier inversion theorem. The Schwartz class. Pointwise convergence of Fourier transforms gives vague convergence in L^1
  
  December 17: Fourier inversion theorem. The Schwartz class. Pointwise convergence of Fourier transforms gives vague convergence in L^1 File