1. Abstract integration: measurable sets and functions. Positive measures.
Integration. Theorems on passage to the limit under the integral sign. Sets of measure zero.
2. The Lebesgue measure in R^n: Pluri-intervals. Inner and outer measure, measurability. Countable additivity and subadditivity.
3. L^p spaces: Holder's and Minkowski's inequalities. Norm. Completeness. Density
of continuous functions with compact support.
4. Hilbert spaces: scalar product, orthogonality. Orthogonal projection onto a closed subspace. Representation of continuous linear functionals. Orthonormal bases. Convergence of trigonometric series.
5. Banach spaces: Bounded linear maps. Baire's Theorem and its consequences. Applications to Fourier series.