Università Roma Tre

Part 1: Introduction to Lebesgue's theory in R^n
Definition of L^1 functions.
Theorems on the integration of limits (monotone convergence, dominated convergence, Fatou's lemma).
Completeness of L^1 (Riesz-Fischer Theorem).
Iterated integrals and Fubini's theorem.
Measurable functions and Lebesgue measure.
Convolution and regularization.
Theorem of the change of variables in R^n.
Divergence theorem in R^n.

Part 2: Fourier in L^2
The Hilbert space L^2 (on bounded domains and on R^n).
Fourier series and transforms in L^2.

Part 3: Fundamentals of the theory of ordinary differential equations
Examples and classes of ordinary differential equations.
Local existence and uniqueness theorem (Picard-Lindelof).
Lipschitz dependence on initial data.
Maximum and global solutions; globality criteria.
Linear systems (linear structure, Wronskian); non-homogeneous systems (variations of constants; Liouville's theorem.
Linear systems with constant coefficients (exponential solution).
Jordan canonical form and qualitative analysis of solutions).
Flows. Variational equation. Parameter dependence C^k.
Introduction to qualitative analysis.
Phase space.
Use of Fourier theory in differential equations (outline).

During the lessons, typed notes will be provided.