GOALS AND CONTENTS

This course aims to provide the students with the knowledge of the basics of some theories of Mathematical Analysis. Such theories play an essential role in the study of Probability and Statistics (measure theory and integration) as well as in the modern approach to partial differential equations (Banach and Hilbert spaces, linear operators). The course has a cultural nature and its further goal is to get the students familiar not only with the results and their applications, but also with the technicalities of the proofs and the check of the hypotheses  through examples and counterexamples.  

TOPICS

0. Basic notions of set theory. Equivalence and order relations. Cardinal and ordinal numbers. Axiom of choice. Topological spaces. Metric spaces. Continuous and semicontinuous functions.

1. Measure space and measurable functions. Positive measures and measurable spaces. Lebesgue measure. Abstract integration. Comparison between Lebesgue and Riemann integrals. Convergences. Derivative of a measure and Radon-Nikodym theorem. Fundamental theorems of Calculus. Product measures and Fubini-Tonelli theorem.  

2. Normed spaces and Banach spaces. Spaces of integrable functions. Linear operators. Dual spaces and Hahn-Banach theorem. Weak convergences. Compact operators.  

3. Hilbert spaces. Scalar product and its consequences. Riesz representation theorem. Orthonormal basis. L^2 space and Fourier series. Spectral theorem for compact symmetric operators. Fredholm alternative. Unbounded linear operators with dense domain.

The exam consists of a written test (compulsory) and an oral examination.