This course explores some of the main ideas and basic working tools of modern 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 the course 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 formulate and communicate the topics of lectures effectively using the appropriate scientific lexicon;
5. be able to read autonomously scientific books and articles on the subject.
Solid knowledge of single and multivariable differential and integral calculus, linear algebra and topology.
Course contents summary
Basic elements of Lebesgue's theory of measure and integration and linear functional analysis in Banach and Hilbert spaces.
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, separabilty, compactness and duals).
9) Lp spaces (completeness, separabilty, compactness and duals).
10) Hilbert spaces.
11) Fourier series: pointwise convergence and L2 theory.
12) Locally convex spaces.
13) Weak and weak* topologies.
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-Verlag, New York 1975;
H. Brezis, "Functional analysis, Sobolev spaces and partial differential equations", Springer-Verlag, New York 2011.
In-person instruction (6 hours per week) and assignments. Students must comply with the university's safety guidelines to mitigate the spread of covid-19. Lectures will be recorded and will be available on the Elly-SMFI web page throughout the semester.
Assessment methods and criteria
Assessment is based on assignments and final oral exam. The final exam will be in-person or online depending on the evolution of covid-19 pandemic.