Real Analysis
A.Y. 2024/2025
Learning objectives
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
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1. Differentiation and integration: Review of the Lebegue integral. Integral functions and the Lebesgue Differentiation 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. Normal operators and compact operators. Spectral Theorem for compact normal operators in Hilbert spaces.
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. Normal operators and compact operators. Spectral Theorem for compact normal operators in Hilbert spaces.
Prerequisites for admission
Knowledge of measure theory and integration including Lebesgue and Hausdorff measures. These arguments are covered in Mathematical Analysis 4.
Teaching methods
In order to facilitate the learning process, classroom lectures and problem sessions are offered to the students, who are strongly encouraged to participate in these activities.
Teaching Resources
- G. Molteni, Note del corso, available in the web site of this course.
- R.L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis, Monographs and Textbooks in Pure and Applied Mathematics, Vol 43, Marcel Dekker, Inc., New York, 1977.
- G. B. Folland, Real Analysis, Modern Techniques and Their Applications, John Wiley & Sons, Inc, New York, 1999.
- E. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration and Hilbert Spaces, Princeton Lectures in Analysis, Vol. III, Princeton University Press, Princeton, 2005.
- E. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton Lectures in Analysis, Vol. I, Princeton University Press, Princeton, 2003.
- E. H. Leib and M. Loss: Analysis, Second Edition, Graduate Studies in Mathematics, Vol 14, American Mathematical Society, Providence, 2001.
- R.L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis, Monographs and Textbooks in Pure and Applied Mathematics, Vol 43, Marcel Dekker, Inc., New York, 1977.
- G. B. Folland, Real Analysis, Modern Techniques and Their Applications, John Wiley & Sons, Inc, New York, 1999.
- E. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration and Hilbert Spaces, Princeton Lectures in Analysis, Vol. III, Princeton University Press, Princeton, 2005.
- E. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton Lectures in Analysis, Vol. I, Princeton University Press, Princeton, 2003.
- E. H. Leib and M. Loss: Analysis, Second Edition, Graduate Studies in Mathematics, Vol 14, American Mathematical Society, Providence, 2001.
Assessment methods and Criteria
The final examination consists homework assignments (typically three assignments with several 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 is required both to illustrate results presented during the course and 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 her/his ability to apply them.
The final examination is passed if the oral exam is successfully passed. Final mark is given using the numerical range 18-30, and is communicated immediately after the oral examination.
- 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 is required both to illustrate results presented during the course and 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 her/his ability to apply them.
The final examination is passed if the oral exam is successfully passed. Final mark is given using the numerical range 18-30, and is communicated immediately after the oral examination.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 9
Practicals: 24 hours
Lessons: 49 hours
Lessons: 49 hours
Professors:
Molteni Giuseppe, Vesely Libor
Shifts:
Educational website(s)
Professor(s)
Reception:
My office: Dipartimento di Matematica, via Saldini 50, first floor, Room 1044.