ELEMENTS OF FUNCTIONAL ANALYSIS
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- DAMIANO FOSCHI
- Credits
- 6
- 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. 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; 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] 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 final exam consists of two parts:
- a written test, during which the student is asked to solve some exercises; a score of at least 15 points over 30 is required to pass the test;
- an oral examination, consisting in a discussion about the theoretical arguments of the course.
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.
Written tests will be delivered as follows:
- 3 tests during the January-February exam session;
- 2 tests during the June-July exam session;
- 1 test during the September exam session. 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)
