TOPICS IN ADVANCED ANALYSIS
The content of the courses: Mathematical Analysis 1-3, Linear Algebra and Geometry, Geometry 1
- Homework exercises which verify the acquisition of an operational knowledge of the subject and the ability to apply the techniques illustrated in class to produce proofs of statements similar to those seen in in the lectures and to express themselves in rigorous mathematical language. - Final oral exam, devoted to the discussion of the homework exercises and to the proof of one or two theorems seen in class. This part tests in-depth knowledge of the topics covered in class, the ability to express oneself in rigorous mathematical language and to recognize the validity of even sophisticated mathematical reasoning. Each part will be evaluated with a grade out of thirty, and the final grade, if greater than or equal to 18, will be the arithmetic mean of the grades of the 2 parts.
The course aims to deepen the study of 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, on the topics treated in the course. They will have learned a number of techniques of proof which they will be able to use to recognize the validity of sophisticated mathematical reasoning and to prove results related to those described in class. Finally students will be able to express themselves in a rigorous mathematical language.
Introduction to Functional Analysis. Normed and Banach sapce. Examples. Finite dimensional spaces. Lp spaces. The Riesz-Fisher theorem. The Hahn-Bananch theorem and consequences. Reflexivity. The Baire, Uniform Boundedness, Open Mapping and Closed Graph Theorems and applications. Weak topologies and weak and weak* convergenze. Sequential compactness theoresm for the weak topologies. The Banach-Alaoglu theorem. 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. Signed and complex measures: total variation and the Hahn and Lebesque decomposition theorems. The Radon-Nikodym theorem. Duals of the Lp spaces. Lebesque differentiation: differentiation of monotone functions, functions of bounded variation, differentiation of an integral and absolutely continuous functions, characterization of absolutely continuous functions. Convolution in Rn, Minkoswki integral inequality and Young’s Theorem. Regularization kernels. Introduction to Hausdorff measure.
Frontal lectures: 64 hours Theoretical notions are developed in the frontal lessons. Homework exercises of theoretical nature will be assigned and corrected in class to provide students with the ability to apply the general abstract techniques described in class in particular situations.
Office hours: by appointment to be set up either at the end of each lecture or by sending an email to alberto.setti@uninsubria.it