2016/2017

2017/2018

DM270

MAT/05 (MATHEMATICAL ANALYSIS)

DEPARTMENT OF PHYSICS

Fisica teorica

2°

1st semester (02/10/2017 - 19/01/2018)

9

78 lesson hours

Italian

WRITTEN AND ORAL TEST

MORA MARIA GIOVANNA (titolare) - 9 ECTS

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

The aim of the course is to introduce the appropriate tools to formulate problems of Mathematical Analysis 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.

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-Arzelà 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.

Lectures and exercise sessions

H. Brézis: Functional analysis, Sobolev spaces and partial differential equations. Springer, 2011.

W. Rudin: Real and complex Analysis. McGraw-Hill, 1987.

Written and oral exam

Written and oral exam