The course introduces to the basic concepts of measure theory (Lebesgue and abstract) and of modern functional analysis, with particular emphasis on Banach and Hilbert spaces. Whenever possible, abstract results will be presented together with applications to concrete function spaces and problems: the aim is to show how these techniques are useful in the different fields of pure and applied mathematics.
At the end of the course, students must be able to read and understand advanced texts on functional analysis. They must be able to solve problems in the discipline.
Lebesgue measure and integral. Outer measures, abstract integration, integral convergence theorems. Banach spaces and their duals. Theorems of Hahn-Banach, of the closed graph, of the open mapping, of Banach-Steinhaus. Reflexive spaces. Spaces of sequences. Lp and W1,p spaces: functional properties and density/compactness results. Hilbert spaces, Hilbert bases, abstract Fourier series. Weak convergence and weak compactness. Spectral theory for self adjoint, compact operators. Basic notions from the theory of distributions.
|Brezis, Haïm||Analisi funzionale. Teoria e applicazioni||Liguori||1986||8820715015|
|A.N. Kolmogorov, S.V. Fomin||Elementi di teoria delle funzioni e di analisi funzionale (Edizione 4)||MIR||1980||xxxx|
|Kolmogorov, A.; Fomin, S.||Elements of the Theory of Functions and Functional Analysis||Dover Publications||1999||0486406830|
|Haim Brezis||Functional Analysis, Sobolev Spaces and Partial Differential Equations||Springer||2011||0387709134|
Written and oral test.
The written test will be based on the solution of open-form problems. The oral test will require a discussion of the written test and answering some questions proposed in open form.
The aim is to evaluate the skills of the students in proving statements and in solving problems, by employing some of the mathematical machinery and of the techniques studied in the course.
The final grade in a scale from 0 to 30 (best), with a pass mark of 18, is given by the arithmetic average of the marks of the written and of the oral part.