Unit FUNCTIONAL ANALYSIS
- Course
- Mathematics
- Study-unit Code
- 55A00085
- Curriculum
- In all curricula
- Teacher
- Enzo Vitillaro
- Teachers
-
- Enzo Vitillaro
- Hours
- 63 ore - Enzo Vitillaro
- CFU
- 9
- Course Regulation
- Coorte 2022
- Offered
- 2022/23
- Learning activities
- Caratterizzante
- Area
- Formazione teorica avanzata
- Academic discipline
- MAT/05
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- The aim of the course is to give basic properties on linear Functional Analysis and to present problems and applications that come from the physical, biological, chemical and economical phenomena.
- Reference texts
- H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011. xiv+599 pp.
Notes from the teacher. - Educational objectives
- The student should acquire a basic knowledge in functional analysis as well as in Banach spaces theory. The course matter is part of the contents of a standard reformed second level course for Italian master degrees in Mathematics. Even the setting is reformed, and the textbooks used are rich of examples and counterexamples, and therefore seem to be optimal to achieve a good understanding of definitions and statements of theorems.
The course aims at analyzing the basic arguments of functional analysis in Banach spaces, at treating so widespread and comprehensive discipline as taught for years at national and international levels. In this sense, the purpose of the course is to make the students able
- To know the main topics of functional analysis and how to apply them to the natural sciences,
- To own computational skills to solve various exercises,
- To read and understand texts of Functional Analysis,
- To provide themselves a mathematical proof of simple statements, with strong reasoning skills,
- To communicate in Italian the mathematical knowledge acquired in the course, as well as related issues,
- To work in teams, but also in autonomy.
The skills listed above are set out in the framework of the professions related to both a traditional mathematician, and a mathematician oriented to technical and/or industrial activities. - Prerequisites
- To better understand the topics covered in the course the student should know the basic subjects of Mathematical Analysis acquired in any Bachelor Degree in Mathematics, Physics and/or Engineering. In particular, the course aims at making the student familiar with the theories that play a central role in modern mathematics, such as functional analysis in Banach spaces and weak topologies, with their use in applications.
- Teaching methods
- The course is split into traditional lectures, in which several exercises are carried out in class to facilitate the understanding of the course. The essential arguments are summarized in handouts provided by the teacher. The course is divided into 63 hours of theory, together with different examples and counterexamples . In the tutorial service the students will be followed individually by the teacher.
To better understand the topics covered in the course the student should know the basic concepts of Mathematical Analysis acquired in any Bachelor Degree in Mathematics, Physics and/or Engineering. In particular, the course aims at making the student familiar with the theories that play a central role in modern mathematics, such as functional analysis in Banach spaces and weak topologies, with their use in applications.
The lecture attendance, even if not mandatory and compulsory, is strongly recommended for a better understanding of the course. - Other information
- The teacher will distribute educational materials useful for a better understanding of the course, in order to help and to let the students pass easily the exam. As an experiment, the course could be done wholly or partly in English, with the agreement of the students attending it. In any case, the oral exam may be conducted in the English language at the request of the student.
The course is divided in 6 hours per week and its time schedule is available at the web pagehttp://www.dmi.unipg.it/didattica/corsi-di-studio-in-matematica/matematica-magistrale/orario-lezioni
Tutorial service is given in office hours. Customized consulting students, also at the request of the students. - Learning verification modality
- The exam includes a single oral test with the performance of some critical exercises. The oral exam consists of a discussion on three topics one of which divided into several questions and takes about 30 minutes. The oral test is designed to assess the level of knowledge attained by the student on the theoretical contents and on the methodologies of the course (fundamental theorems, definitions, examples and counterexamples). Finally, the oral examination allows the teacher to verify the performance of the student and his/her ability to organize the presentation in autonomy.
The examination timetable is divided into at least 8 exam sessions and the examination dates are available at the web page http://www.dmi.unipg.it/didattica/corsi-di-studio-in-matematica/matematica-magistrale/calendario-esami
For information on support services for students with disabilities and/or DSA visit the page http://www.unipg.it/disabilita-e-dsa - Extended program
- Hilbert spaces: generalities and duality. Normed and Banach spaces: the Hahn-Banach Theorem and applications, reflexive spaces, the uniform boundedness theorem and applications; Theorem of Banach-Steinhaus and applications; Strong and weak convergence and applications; the open mapping and closed graph theorems, with applications. Reflexive Banach spaces. Weak topologies: locally convex topological spaces, duality and weak topologies. Weak and weak star topologies: the Banach-Alaoglu and the Krein-Milman theorems, linear bounded operators and weak topologies. Uniform convex spaces and their geometry; Milman-Pettis Theorem; Several theorems on wek and strong convergence in Lp(A); Riesz representation Theorems; Convolution and regularization: Mollifiers and approximation theorems, Young's Theorems, applications; Ascoli–Arzelà theorem, Kolmogorov–Riesz–Fréchet theorem and Dunford–Pettis theorem.