This course explores some of the main ideas and basic working tools of modern analysis, beginning with Lebesgue's theory of measure and integration and moving on towards topics of linear functional analysis in Banach spaces. Application of these ideas and tools to the study of classical problems in real analysis is emphasized.
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 spaces.
1) Abstract integration. Lebesgue's measure and integrals in R^n
2) Real measures and Radon-Nikodym tehorem.
3) Banach spces and bounded linear operators.
4) Banach spaces of continuous functions. Stone-Weierstrass theorem.
5) L^p spaces and their duals.
6) Hahn-Banach theorem, open mapping theorem and uniform boundedness principle.
7) Hilbert spaces.
8) Fourier series: pointwise convergence and L^2 theory.
9) Locally convex spaces.
10 Weak and weak* topologies.
W. Rudin, "Real and complex analysis", McGraw-Hill Inc., New York 1987;
W. Rudin, "Functional analysis", McGraw--Hill Inc., New York 1991;
G. B. Folland, "Real analysis", J. Wiley & Sons, New York 1999;
E. Hewitt -- K. Stromberg, "Real and abstract analysis", Springer-Verlag, New York 1975.
Lectures and exercise sessions (5 hours per week).