This course presents 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 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.