Mathematical Analysis 5
A.Y. 2025/2026
Learning objectives
The course aims at presenting some fundamental topics of Functional Analysis. The first part will be focused on the study of Lebesgue spaces, namely spaces of p-summable functions. The second part will be devoted to the development of the theory of Hilbert spaces and of the operators defined on them.
Expected learning outcomes
Students will become acquainted with some fundamental concepts of Functional Analysis. Upon completion of the course, they will have acquired the knowledge needed to take advanced courses in several areas of study, such as Mathematical Analysis, Probability, Geometry, Mathematical Physics, Mathematical Finance, and Numerical Analysis.
By the end of the classes, the students will have learnt several key results, will be able to provide rigorous proofs of them, and will have developed the ability to autonomously produce abstract arguments to justify more elementary statements. To complement this theoretical training, they will also become capable of solving problems involving concrete reasoning and explicit computations.
By the end of the classes, the students will have learnt several key results, will be able to provide rigorous proofs of them, and will have developed the ability to autonomously produce abstract arguments to justify more elementary statements. To complement this theoretical training, they will also become capable of solving problems involving concrete reasoning and explicit computations.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1. Lebesgue spaces: definition of Lp space, Hölder and Minkowski inequalities, normed space structure and completeness, separability properties, dual space and Riesz representation theorem for Lp spaces, convolution and approximation in Lp by smooth functions, compactness theorems.
2. Hilbert spaces: definition of Hilbert space and fundamental properties, orthogonality and projection theorems, Riesz representation theorem for Hilbert spaces and Lax-Milgram lemma, orthonormal bases, Bessel inequality and Parseval identity, compact operators and spectral theory.
2. Hilbert spaces: definition of Hilbert space and fundamental properties, orthogonality and projection theorems, Riesz representation theorem for Hilbert spaces and Lax-Milgram lemma, orthonormal bases, Bessel inequality and Parseval identity, compact operators and spectral theory.
Prerequisites for admission
Mathematical Analysis 1, 2, 3 and 4.
Teaching methods
Lectures and problem sessions. Attendance is strongly recommended.
Teaching Resources
The topics of the course are essentially covered by the following textbooks:
- G. Teschl, "Topics in Real Analysis", Graduate Studies in Mathematics, American Mathematical Society (to appear), freely available at www.mat.univie.ac.at/~gerald/ftp/book-ra/
- G. Teschl, "Topics in Linear and Nonlinear Functional Analysis", Graduate Studies in Mathematics, American Mathematical Society (to appear), freely available at www.mat.univie.ac.at/~gerald/ftp/book-fa/
- E.H. Lieb, M. Loss, "Analysis", Second edition, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001.
- E.M. Stein, R. Shakarchi, "Real analysis. Measure theory, integration, and Hilbert spaces", Princeton Lectures in Analysis, Vol. 3, Princeton University Press, Princeton, NJ, 2005.
- H. Brezis, "Functional analysis, Sobolev spaces and partial differential equations", Universitext, Springer, New York, 2011.
- G. Teschl, "Topics in Real Analysis", Graduate Studies in Mathematics, American Mathematical Society (to appear), freely available at www.mat.univie.ac.at/~gerald/ftp/book-ra/
- G. Teschl, "Topics in Linear and Nonlinear Functional Analysis", Graduate Studies in Mathematics, American Mathematical Society (to appear), freely available at www.mat.univie.ac.at/~gerald/ftp/book-fa/
- E.H. Lieb, M. Loss, "Analysis", Second edition, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001.
- E.M. Stein, R. Shakarchi, "Real analysis. Measure theory, integration, and Hilbert spaces", Princeton Lectures in Analysis, Vol. 3, Princeton University Press, Princeton, NJ, 2005.
- H. Brezis, "Functional analysis, Sobolev spaces and partial differential equations", Universitext, Springer, New York, 2011.
Assessment methods and Criteria
The examination consists of a written part and possibly of an oral one, optional in nature or at the discretion of the committee.
The written exam is made up of several questions aimed at assessing the students' understanding of the theoretical aspects of the course and their ability to solve problems.
The oral exam involves a discussion mainly focused on the theoretical part of the course programme. It is taken voluntarily, upon successful completion of the written part, or as a mandatory complement of it, if requested by the teachers.
The written exam is made up of several questions aimed at assessing the students' understanding of the theoretical aspects of the course and their ability to solve problems.
The oral exam involves a discussion mainly focused on the theoretical part of the course programme. It is taken voluntarily, upon successful completion of the written part, or as a mandatory complement of it, if requested by the teachers.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Practicals: 24 hours
Lessons: 36 hours
Lessons: 36 hours
Professors:
Cavalletti Fabio, Cozzi Matteo
Professor(s)