The course aims at introducing the students to some fundamental aspects of Real Analysis, with particular reference to Lebesgue and Hilbert spaces.
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. 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 improve their skills to work in small groups of fellow students.
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.
Prerequisites for admission
Knowledge of measure theory and integration including the theory of Lebesgue and Hausdorff. this material is covered in Mathematical Analysis 4.
In order to facilitate the learning process, classroom lectures and problem sessions will be offered to the students, who are strongly encouraged to participate in these activities.
L'esame si articola in tre prove scritte da svolgere a casa a cui segue una prova orale (se le prove scritte sono superate). La prova scritta richiede la soluzione di esercizi aventi contenuti e difficoltà analoghi a quelli affrontati nelle esercitazioni, ed è volta ad accertare le capacità acquisite a risolvere problemi mediante le tecniche sviluppate durante il corso. La prova orale consiste in un colloquio sugli argomenti a programma, volto prevalentemente ad accertare la conoscenza degli argomenti teorici affrontati nel corso.
Assessment methods and Criteria
The final examination consists homework assignments (tipically three assignments with various exercises) and an oral exam.
- In the homework assignments, the student must solve some problems in the form open-ended questions, with the aim of assessing the student's ability to solve problems regarding the main results and definitions presented in the course. - The oral exam can be taken only if the homework assignments have been performed in a regular manner. In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems regarding the main results and definitions presented in the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.
The final examination is passed if the oral exam is successfully passed. Final marks are given using the numerical range 18-30, and will be communicated immediately after the oral examination.