FUNCTIONAL ANALYSIS
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- DAMIANO FOSCHI
- Credits
- 8
- Curriculum
- TEORICO
- Didactic period
- Primo Semestre
- SSD
- MAT/05
Training objectives
- The aim of the course is to provide students with basic functional analysis tools in order to be able to formulate problems, like minimum problems, in suitable functional spaces and to use notions of convergence with respect to suitable topologies for their solutions. For this purpose, basic notions and properties of linear operator theory on Banach and Hilbert spaces will be presented, together with applications to the study of properties of L^p spaces and theis weak topologies. In particular, at the end of the course, students will know the analytic and the geometric versions of Hahn-Banach's theorem, Baire's Lemma, Banach-Steinhaus' theorem, Open mapping and closed graph theorems, Banach-Alouglu-Bourbaki's theorem, Kakutani's theorem, Riezs representation theorem.
At the end of the course students will be able:
- to understand differences between properties of finite dimensional spaces and properties of infinite dimensional spaces (in terms of completeness, compactness, continuity of linear maps, existence of a linear base...);
- to apply the topological notion of continuity, convergence, compactness, separability in topological spaces and metric spaces;
- to study the continuity and compute the operatorial norm of a linear continuous map between linear normed spaces;
- to verify the uniform boundedness of a subset of a Banach space;
- to discuss compactness properties with respect to weak topologies of a convex subset of a reflexive space;
- to discuss compactness properties with respect to weak* topologies of a bounded subset of a reflexive space;
- to discuss the existence of the weak, or weak*, limit for sequences;
- to manage weak and weak* topologies in L^p spaces;
- to discuss which hypotheses on a space X and on a functional f ensure the existence of a minimum for f on X;
- to translate in mathematical language the expression "in the sense of distributions";
- to understand how to study properties of some functional spaces such as BV and Sobolev spaces;
- apply the direct method of calculus of variations to the study of minimum problems. Prerequisites
- As prerequisites, it is required full comprehension of the basic mathematical analysis and calculus notions of undergraduate level,
in particular good knowledge of measure and integration theory,
and good knowledge of basic general topology. Course programme
- (1) Review of basic topology. Separable spaces. Compact spaces. Compactness in metric spaces. Continuous and semicontinuous functions. Existence of minimum theorems. [4 hours]
(2) Seminormse and norms. Topology and notion of convergence in a normed space. Equivalence of norms. Review of finite dimensional vector spaces: norm equivalence, completeness, Bolzano-Weistrass theorem and compacteness of closed balls. Riesz's theorem. [3 hours]
(3) Banach spaces. Example of non equivalent norms. Weirstrass criteria for the convergence of series in Banach spaces. Examples of infinite dimensional normed spaces:
(a) sequence spaces c_0, c_00, l^p;
(b) function spaces: C^k; Ascoli-Arzelà theorem;
(c) L^p spaces;
[4 hours]
(4) Linear operators between normed spaces: boundedness and continuity. Operatorial norm. Adjoint operator. Completeness of the linear operator space L(X,Y). Duals of l^p spaces.
[4 hours]
(5) Zorn's Lemma, Hamel's base. Real analytic form of Hahn-Banach theorem. Construction of unbounded linear functionals. Extensions of real bounded linear operators defined on linear subspaces of a normed space. Closed hyerplanes. Geometric forms of Hahn-Banach theorem. Separations of convex sets. [8 hours]
(6) Baire's lemma and its equivalent forms. First and second category spaces. Banach-Steinhaus theorem. Bounded sets in X and in X'. Open mapping and closed graph theorems. [7 hours]
(7) Topology induced by a family of functions. product topology. Weak topology and weak* topology. Weak and weak* notions of convergence. Closure of a conves set. Banach-Alaoglu-Bourbaki theorem. Reflexive spaces. Kakutani's theorem. Separable spaces. Uniformly convex spaces. [8 hours]
(8) Reflexivity of Hilbert spaces. Theorem of projections on convex closed subsets in Hilbert spaces. Riesz-Frechet theorem. Lax-Milgram theorem. [3 hours]
(9) L^p spaces: uniform convexity, reflexivity, separability. Duals of L^p spaces. Riesz representation theorems. Caracterization of weak and weak* convergence. Definition of compact operator. [7 hours]
(10) Introduction to Sobolev spaces: definitions and basic properties (completeness, reflexivity). Poincaré inequality. Weakly harmonic functions. Weak solutions of Poisson equation. Existence of weak solutions of Dirichlet problem for Poisson equation by Hilbert space methods. [16 hours] Didactic methods
- The course will be presented through lectures and exercises. Theorems will be introduced and proved in details, together with discussions of examples, applications and exercises and assignments of home exercises.
Learning assessment procedures
- The course learning assessment is done through homework assignments and a final exam. Admission to the final exam is granted only after submission of the homework assignments.
The exam consists of two parts:
- a written examination of 3 hours, where the student is asked to solve some exercises (usually 4)
- an oral examination, consisting in a discussion about the theoretical arguments of the course.
The student is asked to discuss some of the theorems studied and of their applications.
A student is admitted to the oral examination if his mark at the written examination is greater or equal to 15. The oral examination has to be taken in the sames session as the written exam, and before the beginning of the next term's lessons.
The final mark depends on both the examinations. It isn't given by the arithmetical mean between the written and the oral mark, but it comes from an overall evaluation of the student's skill.
In the winter session there are at least 2 possibilities to take both the written and the oral exam, the dates are arranged with the students of the class. In the other sessions there are one or more opportunities to take the exam, depending on the student's demand. Reference texts
- - Lecture notes provided by the teacher.
Specific topics can be found in the following textbooks:
- W. Rudin "Real and complex analysis", McGraw-Hill (1986)
- H. Brezis "Functional Analysis, Sobolev Spaces and Partial Differential Equations" (Springer)
- G. Gilardi "Analisi 3" Mc Graw Hill
- H. Brezis "Analisi funzionale", Liguori editore (1990)
