TOPICS IN ADVANCED ANALYSIS A
The content of the following courses: Mathematical Analysis 1, 2, and 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. ommunication 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.
The course is divided into two parts: A first part dedicated to foundational topics in real and functional analysis, and a second part focused on more advanced topics. Part 1: Introduction to Functional Analysis Normed spaces and Banach spaces. Examples. Finite-dimensional spaces. Lp spaces. Riesz–Fischer Theorem. Hahn–Banach Theorem and 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. Trigonometric polynomials and L2-theory of Fourier series: Bessel's inequality, Parseval’s identity, and Plancherel’s identity. Orthonormal bases in L2(−π,π). Convolution in RnRn, Minkowski’s integral inequality, and Young’s inequality. Regularizing kernels. Part 2: Depending on the composition of the class and the time available, topics will be selected from the following list: Geometric form of the Hahn–Banach Theorem. Strong and weak closures of convex sets. Mazur’s Lemma. Extreme points and the Krein–Milman Theorem. Reflexivity: Kakutani and Eberlein–Šmulian Theorems. Uniformly convex spaces. Milman–Pettis Theorem. Uniform convexity of Lp spaces. Topological vector spaces and seminormed spaces. Metrizability. Topology of spaces of differentiable functions and introduction to distributions. Linear operators and their adjoints. Projection, isometric, and unitary operators. Finite-rank and compact operators. Hilbert–Schmidt operators. Closed and closable operators. Symmetric, self-adjoint, and normal operators. Self-adjoint extensions of symmetric operators. Basic notions of spectral theory. Spectral theory of compact operators. Spectral theorem for self-adjoint and normal operators. Introduction to the theory of semigroups of operators. Semigroups and their generators. Hille–Yosida Theorem. Applications to differential equations. Further topics in measure theory: Construction of measures. Product measures. Tonelli–Fubini Theorem. Positive Borel measures. Riesz Representation Theorem. Continuity properties of measurable functions. Signed and complex measures: Total variation, Hahn and Lebesgue decomposition theorems. Radon–Nikodym Theorem. Characterization of the dual of Lp spaces. Hardy–Littlewood maximal function. Differentiation of measures. Lebesgue differentiation: Differentiation of monotone functions, functions of bounded variation, differentiation of an integral, and absolutely continuous functions. Characterization theorem for absolutely continuous functions. Introduction to interpolation: Riesz–Thorin and Marcinkiewicz Interpolation Theorems. Lp spaces: Density and separability properties. Compactness in Lp: Kolmogorov–Riesz Theorem. Additional topics on Fourier series: Dirichlet and Fejér kernels. Hausdorff–Young Theorem. Pointwise convergence, Dini’s Theorem, and absolute convergence, Bernstein’s Theorem. Fourier transform in Rn: L1 theory and inversion theorem; L2 theory: Plancherel’s identity and inversion theorem. Hausdorff–Young inequality.
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.
Office hours: By appointment, to be arranged either at the end of a lecture or by emailing alberto.setti@uninsubria.it.
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Degree course in: MATHEMATICS