Timetable
- October 19, 14.30-16.30: basic notions of linear algebra. Definition of measure space. Borel sigma-algebra on R.
- October 26, 14.30-16.30: characterization of sigma-finite borelian measures on R in terms of the cumulative distribution function. The Lebesgue measure. The counting measure. Completion of a sigma-algebra with respect to a measure, definition of the sigma-algebra of Lebesgue measurable sets. Measurable functions and random variables.
- October 27, 10-12: Measurable functions and associated distribution. Absolutely continuous and singular measures with respect to Lebesgue measure. Examples. Radon-Nikodym theorem. Lebesgue integral. Characterization of absolutely continuous sigma finite measures in terms of their density function.
- November 2, 14.30-16.30: Moments of random variables. Normed spaces, metric structure induced by the norm, Cauchy sequences, Banach spaces. Banach-Caccioppoli fixed point theorem. Definition of possible norms in L^p spaces and in spaces of random variable with finite p-moment, M_p. Young inequality.
- November 3, 10-12: Holder and Minkowski inequalities. Convergence in p-mean. Jensen’s inequality. Random variables with bounded p-moment have also bounded q-moment, for every q<p. Bounded linear operators in Banach spaces.
- November 9, 14.30-16.30: bounded linear operators between Banach spaces. Norm of a bounded linear operator. Example of linear bounded operator from M_p to R (Riesz representation theorem – hints). Definition of scalar product and of Hilbert space. Examples. Definition of orthogonal elements and of orthogonal spaces.
- November 10, 10-12: Example of a linear bounded operator from M_2 to istself (conditional expectation operator). Orthogonal projection theorem in Hilbert spaces. Definition of orthonormal set and of orthonormal basis. Characterization of the orthogonal projection on a closed subspace V of H in terms of a orthonormal basis of V.
- November 17, 10-12: example of computation of orthogonal projection:linear least square estimator, adjoint of a linear bounded operator of a Hilbert space, definition of compact operator, definition of eigenvalues of an operator (point spectrum), spectral decomposition theorem for linear compact symmetric operators in Hilbert spaces.
Exam (written examination): December 13, 10-11.30
Bibliography
G. B. Folland Real Analysis: modern tecniques and their applications. Wiley 1999 (2nd ed)
A. Bressan Lecture notes on Functional Analysis, Graduate Studies in Mathematics AMS 2013
Past exam papers, with sketch of solutions