Students should be able to independently produce proofs of elementary statements (aided also by the experience obtained through homework assignments), and be able to explain rigorously the theoretical knowledge and computational aspects learned from the lectures and assigned problems. Moreover they will have the opportunity to work in small groups of fellow students.
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 spaces.
Lesson period: First semester
(In case of multiple editions, please check the period, as it may vary)
1. Differentiation and integration: Review of the Lebegue integral. Integral functions and the Lebesgue Differentiation Theorem. Signed measures and the Radon-Nikodym Theorem. Differentiation of monotone functions. Functions of bounded variation, Absolutely continuous functions. Convex functions and Jensen's inequality.
2. Lp Spaces: Definition, Hölder and Minkowski inequalities, convergence and completeness. Comparison of notions of convergence. The dual space of Lp and the Riesz Representation theorem for Lp. Convolution and the inequalities of Young. Approximation in Lp by regular functions.
3. Hilbert spaces: Definition and fundamental properties. The Projection Theorem. Orthonormal bases. Continuous linear functionals and the Riesz Representation Theorem for Hilbert Spaces. Bilinear forms and the Lax-Milgram Theorem. Orthonormal bases and separability. Expansions in Fourier series and fundamental examples of complete orthonormal systems. Convolutions kernels and pointwise convergence of Fourier series. Bounded linear operators. Self-adjoint operators and compact operators. Spectral Theorem for compact self-adjoint operators.