The aim of the course consists in introducing students to fundamental questions of the modern Real Analysis paying particular attention to how these concepts can be used to solve both integral an differential equations.
Expected learning outcomes
Students will acquire familiarity with the basic properties of the real analysis, with particular focus on the theory of Lebesgue and Hilbert' s spaces. They will be able to produce autonomously proofs of simple statements (homeworks), and to explain rigorously the knowledges and connected problems learnt in the lectures. Moreover they will have the opportunity of working in groups.
1. Differentiation and integration Integral functions, Lebesgue's Differentiation Theorem (TDL). Differentiation of monotone functions. Functions of bounded variation, Absolutely continuous functions. Convex functions and Jensen's inequality.
2. Lp Classes: Definition, Hölder and Minkowski inequalities, Riesz-Fischer Theorem, weak and strong convergence in Lp. the dual space of Lp. convolution and Young inequalities. Approximation in Lp by regular functions.
3. Hilbert spaces: Definition and fundamental properties. The Projection Theorem. Orthonormal bases. Continous linear functionals, Riesz' s Theorem. Bilinear forms and the Lax-Milgram Theorem.
4. Elements of Fourier analysis: Periodic functions and Fourier series; pointwise convergence and convergence in L2. The Fourier Trasform, the inversion Theorem.