The aim of the course is to introduce the appropriate tools to formulate Mathematical Analysis problems in spaces of infinite dimension. The fundamental results of Functional Analysis will be discussed, with a focus on the theory of Banach and Hilbert spaces.

Lectures and exercise sessions

Written and oral exam.

Multivariable differential and integral calculus. Lebesgue measure and integration. Basic notions of linear algebra.

Norms and scalar products. Topological vector spaces. Normed spaces. Bounded linear operators. Topological dual space.

Banach spaces. Hahn-Banach Theorem: analytical and geometrical forms and their consequences. Baire Lemma. Banach-Steinhaus Theorem. Open Mapping Theorem, Closed Graph Theorem, and their consequences.

Weak* topology, weak topology, and their properties. Banach-Alaoglu Theorem. Reflexive spaces. Separable spaces.

L^p spaces. Elementary properties. Reflexivity and separability of L^p. Riesz Representation Theorem. Approximation by convolution. Ascoli-Arzelŕ Theorem. Fréchet-Kolmogorov Theorem.

Hilbert spaces. Projection on a convex closed set. Riesz Representation Theorem for the dual space. Stampacchia Theorem. Lax-Milgram Theorem. Complete orthonormal systems.

Compact operators. Adjoint of a bounded operator. The Fredholm Alternative. Spectrum of a compact operator. Spectral decomposition of a compact self-adjoint operator. Integral operators. Application to Sturm-Liouville problems.

H. Brézis: Functional analysis, Sobolev spaces and partial differential equations. Springer, 2011.
W. Rudin: Real and complex Analysis. McGraw-Hill, 1987.