ISTITUZIONI DI ANALISI SUPERIORE
The content of the courses Mathematical Analysis 1-3, Linear Algebra and Geometry, Geometry 1
Aims and outcomes
The course is the natural continuation of the sequence Mathematical Analysis 1-3, and it aims to deepen the study of classical and modern analysis begun in the previous courses.
Students will acquire a working knowledge of the methods of advanced analysis. They will know statements and proofs of the main theorems, and will be able to solve exercises, even of theoretical nature, about the topics dealt with in the course. They will have learned a number of techniques of proof which they will be able to use to prove results related to those described in class.
Program
Hilbert spaces. Orthogonality and orthonormal bases. Riesz representation theorem and Hilbert space duals. Orthonormal bases in L2(-\pi,\pi). Trigonometric polynomials and Fourier series on the torus. L2
theory: Bessel inequality and Parseval and Plancherel identities Polinomi trigonometrici e serie di Fourier sul toro. Teoria L2: Bessel inequality and Parseval and Plancherel identities. Pointwise convergenge. Isoperimetric inequality in R2.
Lebesque differentiation: differentiation of monotone functions, functions of bounded variation, differentiation of an integral and absolutely continuous functions, characterization of absolutely continuous functions.
Topics in measure theory: Borel measure and regularity property. The Riesz representation theorem. Luzin theorem. Signed and complex measures: total variation and the Hahn and Lebesque decomposition theorems. The Radon-Nykodym theorem. Duals of the Lp spaces.
Convolution in Rn, Minkoswki integral inequality and Young’s Theorem. Regularization kernels. Introduction to Hausdorff measure.
Teaching methods
Frontal lectures: 64 hours
Textbooks
- E. Giusti, Analisi Matematica 2, Boringhieri
- H. Royden, Real Analysis, Mc Millan
- W. Rudin, Real and Complex Analysis, Mc Graw Hill.
Final examination
Written and oral examination
Borrowed from
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