ID:
SCC1145
Duration (hours):
72
CFU:
9
SSD:
ANALISI MATEMATICA
Year:
2025
Overview
Date/time interval
Secondo Semestre (23/02/2026 - 12/06/2026)
Syllabus
Course Objectives
Knowledge and understanding:
The student will acquire an operational knowledge of advanced analysis methods, building upon concepts learned in previous courses. They will become 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.
The student will acquire an operational knowledge of advanced analysis methods, building upon concepts learned in previous courses. They will become 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.
Course Prerequisites
The content of the courses in mathematical analysis, linear algebra, geometry, and general topology typically offered in a three-year bachelor's degree program in mathematics.
Teaching Methods
Lectures: 72 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, as well as to apply the general abstract techniques described in class in particular cases.
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, as well as to apply the general abstract techniques described in class in particular cases.
Assessment Methods
- 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.
- 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.
Contents
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's Theorem, Uniform Boundedness Principle, Open Mapping Theorem, Closed Graph Theorem, and applications. Weak topologies, weak convergence, and weak-star convergence. Sequential compactness theorems in weak topologies. Banach-Alaoglu Theorem.
Hilbert spaces. Orthogonality and orthonormal bases. Riesz Representation Theorem and the dual of Hilbert spaces.
Complements of Measure Theory: Construction of measures. Product measures. Tonelli-Fubini Theorem. Signed measures and complex measures: Total variation, Hahn and Lebesgue decomposition theorems. Radon-Nikodym Theorem. Characterization of the dual of Lp spaces. Hardy-Littlewood maximal function. Lebesgue differentiation: differentiation of monotone functions, functions of bounded variation, differentiation of an integral, absolutely continuous functions, characterization theorem for absolutely continuous functions.
Convolution in Rn, Minkowski's integral inequality, and Young's theorem. Regularizing kernels. Introduction to Hausdorff measure.
Fourier Series: Orthonormal bases in L2(-π, π). Trigonometric polynomials and Fourier series on the torus. L2 theory: Bessel's inequality, Parseval's identity, and Plancherel's theorem. Pointwise convergence: Dirichlet kernel. Riemann-Lebesgue lemma. Dini's theorem. Absolute convergence: Bernstein's theorem.
Depending on the composition of the class, monographic topics will also be selected from among: theory of semigroups of operators; introduction to approximation theory; topological vector spaces, distributions, and fundamental solutions of differential equations with constant coefficients; introduction to the theory of singular integrals; Banach algebras and spectral theory.
Hilbert spaces. Orthogonality and orthonormal bases. Riesz Representation Theorem and the dual of Hilbert spaces.
Complements of Measure Theory: Construction of measures. Product measures. Tonelli-Fubini Theorem. Signed measures and complex measures: Total variation, Hahn and Lebesgue decomposition theorems. Radon-Nikodym Theorem. Characterization of the dual of Lp spaces. Hardy-Littlewood maximal function. Lebesgue differentiation: differentiation of monotone functions, functions of bounded variation, differentiation of an integral, absolutely continuous functions, characterization theorem for absolutely continuous functions.
Convolution in Rn, Minkowski's integral inequality, and Young's theorem. Regularizing kernels. Introduction to Hausdorff measure.
Fourier Series: Orthonormal bases in L2(-π, π). Trigonometric polynomials and Fourier series on the torus. L2 theory: Bessel's inequality, Parseval's identity, and Plancherel's theorem. Pointwise convergence: Dirichlet kernel. Riemann-Lebesgue lemma. Dini's theorem. Absolute convergence: Bernstein's theorem.
Depending on the composition of the class, monographic topics will also be selected from among: theory of semigroups of operators; introduction to approximation theory; topological vector spaces, distributions, and fundamental solutions of differential equations with constant coefficients; introduction to the theory of singular integrals; Banach algebras and spectral theory.
Course Language
English
More information
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
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MATHEMATICS
Master’s Degree
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