Course of the III year of Laurea Triennale in Mathematics; the course consists of two parts: complex analysis and functional analysis, both at introductory level.
The objective of the course is to provide an introduction to three areas of Mathematical Analysis which are fundamental for their theoretical and applied implications and developments. Complex analysis: Cauchy-Riemann equations, Cauchy theorem and consequences, singularities and Laurent series, residue theorem. Hilbert spaces: normed spaces and bounded linear operators, Hilbert spaces and orthonormal bases, L1 and L2 spaces, Riesz representation and projection theorems. Fourier series: Fourier series in L ^ 2, L ^ 1, pointwise convergence.
Students will be able to solve simple problems and to follow more advanced studies in complex and functional analysis.
Analysis, Geometry and Algebra courses from the first two years of Laurea Triennale in Mathematics.
The classical method: lectures and exercises using the blackboard; written (exercises) and oral (theory and exercises) examination. Special importance is given to exercises and consequently to the written part of the exam.
Complex Analysis: power series and analytic functions; complex differentiation and holomorpgic functions; complex integration, Cauchy's theorem and primitives; classical consequences of Cauchy's theorem; singularities, residue theorem and applications.
Functional Analysis: normed spaces; linear operators; scalar products; Hilbert spaces and orthonormal bases; projection theorem and Riesz representation theorem; study of important examples: the space L^2.
Fourier Analysis: Fourier series in L^2. Fourier series in L^1: an outline of results on the convergence, punctual and in norm.
V.Villani - Funzioni di Una Variabile Complessa - Edizioni Scientifiche Genova 1971.
I.Stewart, D.Tall - Complex Analysis, 2nd ed. - Cambridge U. P. 2018.
H.Cartan - Elementary Theory of Analytic Functions of One or Several Variables - Dover Publ. 1995.
A.I.Markushevich - Theory of Functions of a Complex Variable, parts I--III - A.M.S. Chelsea Publishing 2005.
W.Rudin - Analisi Reale e Complessa - Bollati Boringhieri 1978.
M.Reed, B.Simon - Functional analysis - Academic Press 1972.
E.M.Stein, R.Shakarchi - Real Analysis - Princeton U. P. 2005.
Ricevimento: On appointment; take directly an appointment with the professor or write to perelli@dima.unige.it
When lessons of the III year start.
Written and oral examination.
Students with a certified DSA, disability or other special educational needs are advised to contact the lecturer at the beginning of the course in order to agree on teaching and examination methods that, while respecting the teaching objectives, take into account individual learning methods and provide suitable compensatory tools.
Evaluation of written and oral examination. In the written part, some exercises will be proposed, and the quality of the solutions written by the students will be evaluated. The oral part deals mainly with the theory developed during the course, and the understanding of the theorems and the ability of reproducing proofs of the students will be evaluated.
