TOPICS IN ADVANCED ANALYSIS
The content of the courses: Mathematical Analysis 1-3, Linear Algebra and Geometry, Geometry 1
- Homework exercises aimed at assessing the acquisition of an operational understanding of the subject, the ability to express oneself using rigorous mathematical language, and the capacity to independently produce proofs similar to those presented in class by applying the techniques illustrated during the lectures. - Final oral exam dedicated to the discussion of the completed exercises and the proof of selected theorems covered in class. This part assesses an in-depth knowledge of the course topics, the ability to express oneself in rigorous mathematical language, and the capacity to recognize the validity of even sophisticated mathematical reasoning. Each part will be graded on a 30-point scale. The final grade, if equal to or greater than 18, will be the arithmetic mean of the two components.
Knowledge and understanding: The student will acquire an operational knowledge of advanced analysis methods, building upon concepts learned in previous courses. They will be familiar with the main theoretical statements and their proofs, developing a solid and rigorous understanding of the foundations of modern analysis. Applying knowledge and understanding: The student will be able to apply the acquired knowledge to solve exercises, including those of a theoretical and abstract nature, related to the topics covered in the course. They will also be capable of using these skills to analyze advanced mathematical problems. Making judgements: The course will provide the student with a repertoire of proof techniques that will enable them to independently assess the validity of mathematical reasoning, even in complex contexts, and to construct rigorous proofs of results related to those presented in class. Communication skills: The student will be able to express themselves with precision and clarity in the field of mathematics, using a formal and rigorous language suitable for conveying complex mathematical ideas. Learning skills: The theoretical and structured approach of the course will help the student develop independent study and in-depth learning abilities, laying the groundwork for further learning in analysis and mathematics at a higher level.
Introduzione all'analisi funzionale. Spazi normati e spazi di Banach. Esempi. Spazi di dimensione finita. Spazi Lp. Teorema di Riesz-Fisher. Teorema di Hahn-Banach e conseguenze. Riflessività. Teoremi di Baire, di Uniforme Limitatezza, dell'Applicazione Aperta e del Grafico Chiuso e applicazioni. Topologie deboli e convergenza debole e debole stella. Teoremi di compattezza per successione nelle topologie deboli. Teorema di Banach-Alaoglu. Spazi di Hilbert. Perpendicolarità, e basi ortonormali. Teorema di rappresentazione di Riesz e duale dgli spazi di Hilbert. Complementi di teoria della misura: Costruzione di misure. Misure prodotto. Il teorema di Tonelli-Fubini. Misure con segno e misure complesse: Variazione assoluta, e teoremi di decomposizione di Hahn e di Lebesgue. Teorema di Radon-Nikodym. Caratterizzazione del duale degli Lp. La funzione massimale di Hardy-Littlewood. Differenziazione di Lebesgue: differenziazione di funzioni monotone, funzioni a variazione limitata, differenziazione di un integrale, e funzioni assolutamente continue, teorema di caratterizzazione delle funzioni assolutamente continue. Convoluzione in Rn, disuguaglianza integrale di Minkowski e teorema di Young. Nuclei regolarizzanti. Introduzione alla misura di Hausdorff. Serie di Fourier: Basi ortonormali in L2(-\pi,\pi). Polinomi trigonometrici e serie di Fourier sul toro. Teoria L2: disuguaglianza di Bessel e identità di Parseval e Plancherel. Convergenza puntuale: nucleo di Dirichlet. Lemma di Riemann Lebesque. Teorema di Dini. Convergenza assolute: teorema di Bernstein. Introduction to Functional Analysis: Normed spaces and Banach spaces. Examples. Finite-dimensional spaces. Lp spaces. Riesz–Fischer Theorem. Hahn–Banach Theorem and its consequences. Reflexivity. Baire Category Theorem, Uniform Boundedness Principle, Open Mapping Theorem, and Closed Graph Theorem, with applications. Weak topologies and weak and weak-* convergence. Compactness theorems for sequences in weak topologies. Banach–Alaoglu Theorem. Hilbert spaces: Orthogonality and orthonormal bases. Riesz Representation Theorem and the dual of Hilbert spaces. Additional Topics in Measure Theory: Construction of measures. Product measures. Tonelli–Fubini Theorem. Signed and complex measures: Total variation, Hahn and Lebesgue decomposition theorems. Radon–Nikodym Theorem. Characterization of the dual of Lp spaces. The Hardy–Littlewood maximal function. Lebesgue Differentiation of measures. Differentiation of monotone functions, functions of bounded variation, differentiation of an integral, and absolutely continuous functions. Characterization theorem for absolutely continuous functions. Convolution in Rn: Minkowski's integral inequality and Young’s Theorem. Regularizing kernels. Fourier Series: Orthonormal bases in L2(−π,π)L2(−π,π). Trigonometric polynomials and Fourier series on the torus. L2L2-theory: Bessel’s inequality, Parseval’s identity, and Plancherel’s identity. Pointwise convergence: Dirichlet kernel. Riemann–Lebesgue Lemma. Dini’s Theorem. Absolute convergence: Bernstein’s Theorem.
Lectures: 64 hours During the lectures, theoretical concepts are developed. Theoretical exercises assigned as homework and corrected in class will provide students with the ability to apply the general proof techniques presented during the lectures to specific situations.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