FUNCTIONAL ANALYSIS
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- Versione italiana
- Academic year
- 2020/2021
- Teacher
- FRANCESCA AGNESE PRINARI
- Credits
- 9
- Didactic period
- Primo Semestre
- SSD
- MAT/05
Training objectives
- The intent of the lectures is to provide the student with an introduction to Functional Analysis that not only presents the basic notions, the theorems and the techniques, but also gives a sampling of the applications. We introduce the Banach spaces and the Hilbert spaces, we characterize the continuity of linear applications on them, we show the main theorems of the Functional Analysis (Hahn-Banach Theorems, Baire Lemma, Banach-Steinhaus Theorem, Open Mapping Theorem, Closed Graph Theorem, Banach-Alouglu-Bourbaki Theorem, Kakutani Theorem, Rietz representation theorem).
Moreover we introduce the main properties of the weak topologies in order to show the existence of minimizers for a suitable class of functionals defined on reflexive Banach spaces. At the end of the course the student will be able to
-compare a finite dimensional space with an infinite dimensionale space and discuss their difference (with respect to the property of completeness, compactness, continuity of linear functionals, existence of basis..)
-to apply the notions of continuity, convergence, compactness, separability in topological spaces (not only in metric spaces)
-to compute the norm of a linear and continuous functional
-to discuss the uniform boundedness of a subset of a Banach space
-to discuss the compactness of a convex subset with respect to the weak topology in a reflexive space
-to discuss the weak or weak* convergence of a sequence
- to make use of the weak in L^p and of weak* topology in L^\infty
to discuss the existence of minimizers for some classes of minimization problems
-to undertand the meaning of the expression ''in the sense of distribution ''
-to approach the study of further spaces as BV and the Sobolev spaces
-to apply the direct methods of Calculus of Variations in order to study minimization problems. Prerequisites
- The main items required are some knowledge of point set topology together with a good background in measure and integration theory.
Course programme
- (1) Topological spaces. Basis, sub-basis, fundamental system of neighborhoods. Hausdorff spaces. First and Second-countable spaces. Separable spaces. Compact spaces. Sequential compactness. Compact metric spaces. Lower semicontinuity. Generalized Weirstrass theorems. Product of topological spaces. (4 hours)
(2)Normed spaces. Equivalent norms. Finite dimensional normed spaces: completeness with respect to any norm. Riesz' Lemma. (3 hours)
(3)Example of Banach spaces: c_0, c_00, l^p, C(K), C^1(K), L^p. Ascoli Arzela's Theorem (4 hours).
(4)Linear and continuous functionals. Norm of a linear functional. The dual space. Examples. (4 hours)
(5) Zorn's Lemma, Hamel basis. The Analytic Form of the Hahn–Banach Theorem: Extension of Linear Functionals (the real case). Closed hyperplane.
The Geometric Forms of the Hahn–Banach Theorem:
Separation of Convex Sets. (8 hours)
(6)The Baire Category Theorem; Banach-Steinhaus-Theorem: corollaries and applications. The Open Mapping Theorem and the Closed Graph Theorem. (7 hours)
(7)Weak Topologies. Reflexive Spaces. Separable
Spaces. Uniform Convexity. (8 hours)
(8)Reflexivity of Hilbert spaces. Projection. The dual space. Riesz–Fréchet representation theorem. Lax-Milgram's Theorem (3 hours).
(9)L^p spaces: reflexivity when 1 < p < +\infty and separability when 1 = p < +\infty. Dual of L^p: Riesz representation theorem. Weak and weak* convergence. Definition of compact operator. (6 hours)
(10) Sobolev spaces: definition and main properties. Sobolev Imbedding in $W^1,p_0(\Omega)$. (16 hours) Didactic methods
- The course includes theoretical lectures, accompanied by exercises. The main theorems are shown and examples and applications are provided. Moreover a lot of exercises are proposed to the students in order to train. Finally the students could prepare for the exam by solving the previous written examinations which are available (with their solutions) on the website of the course.
Learning assessment procedures
- The examinations is divided in two parts: a written partial exam (which requires the resolution of some exercises) and an oral examination. To pass the written test it is required to get at least 16 points out of 30. The final mark V takes into account both the partial scores according to the formula:
V=(2m+3 M):5, rounded up to the nearest integer,
where m=minimum{O,S}, M=maximum{O,S} and O=mark of the oral test and S=mark of the written test.
It is possibile to split the written partial exams in two parts. The first test takes place after the first half part of the course, the second one at the end of the course. After passing the written test, the student can take the oral test within the first data of the next session of exams. Reference texts
- -Teacher’s handouts
Specific topics can be further developed in the following texts:
-W.Rudin "Real and complex analysis", McGraw-Hill (1986)
-H. Brezis ''Functional Analysis, Sobolev Spaces and Partial Dierential Equations'' (Springer)
-G. Gilardi ''Analisi 3'' Mc Graw Hill
-H. Brezis "Analisi funzionale", Liguori editore (1990)
