Learning objectives
This course presents some of the main ideas and basic working tools of modern mathematical analysis, starting with Lebesgue's theory of measure and integration and moving on towards topics of linear functional analysis in Banach and Hilbert spaces including weak topologies. Applications to the study of classical problems in real analysis are emphasized.
By the end of lectures students must
1. exhibit solid knowledge and thorough conceptual understanding of the subject;
2. be able to produce rigorous proofs of results related to those examined in the lectures;
3. be able to evaluate coherence and correctness of results obtained by themselves or by others;
4. be able to communicate the course content effectively using the appropriate scientific lexicon;
5. be able to read autonomously scientific books and articles on the subject.
Prerequisites
Solid knowledge of single and multivariable differential and integral calculus, linear algebra and topology.
Course unit content
Basic elements of Lebesgue's theory of measure and integration and linear functional analysis in Banach and Hilbert spaces.
Full programme
1) Measure theory and abstract integration.
2) Caratheodory's construction of measures.
3) Lebesgue's measure and integral in R^n
4) Locally compact Hausdorff spaces and Radon measures.
5) Real/complex measures and Radon-Nikodym theorem.
6) Banach spaces and bounded linear operators. Dual space.
7) Hahn-Banach theorem, open mapping theorem and uniform boundedness principle.
8) Banach spaces of continuous functions (completeness, separability, compactness and duals).
9) Lp spaces (completeness, separability, compactness and duals).
10) Hilbert spaces.
11) Spectral theory for compact, selfadjoint operators.
12) Fourier series: pointwise convergence and L2 theory.
13) Weak and weak* topologies.
Bibliography
Handouts and material taken from the following textbooks:
W. Rudin "Real and complex analysis", McGraw-Hill, New York 1987
W. Rudin "Functional analysis", McGraw-Hill, New York 1991
G. B. Folland "Real analysis. Modern techniques and applications", J. Wiley & Sons, New York 1999
E. Hewitt -- K. Stromberg "Real and abstract analysis", Springer, New York 1975
D. Cohn "Measure theory", Birkhäuser/Springer, New York 2013
Teaching methods
In-person instruction (6 hours per week) and assignments. Attendance at classes is strongly recommended.
Assessment methods and criteria
Assessment is based on assignments and an oral examination. The oral examination aims at assessing knowledge and comprehension of the contents of lectures.
Exams will be in-person.
Other information
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2030 agenda goals for sustainable development
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