Real and Functional Analysis
Lecture by lecture, the course of Real and Functional Analysis at Politecnico di Milano: measure theory and integration first, then infinite-dimensional vector spaces.
- Cardinality, topology and measurable spaces
- Measure spaces, completeness and outer measures
- Lebesgue measure, the Cantor set and the Cantor function
- The Vitali example, measurable functions and simple functions
- The Lebesgue integral, vanishing lemmas and the Riemann integral
- Fatou's lemma, monotone and dominated convergence, and the spaces
- Notions of convergence of measurable functions
- Exercises on limits of integrals and notions of convergence
- Differentiation of measures, the fundamental theorems of calculus and Sobolev spaces
- Banach spaces, the Hölder inequality and the completeness of
- Exercises on absolute continuity, weak derivatives and convergence in
- Equivalent norms, density and compactness in infinite dimensions