This course will provide the foundations of Modern Analysis: measure theory, Hilbert spaces, spectral theory of self-adjoint operators. We will also discuss applications to differential and integral equations.

Detailed syllabus:

Measure, exterior measure, Lebesgue measure, Borel sets, measurable functions, integration, convergence in measure, almost everywhere convergence, Lebesgue convergence theorems, completeness of Lp spaces, absolutely continuous functions, comparison with Riemann integration, product measures, Fubini's theorem.

Hilbert spaces: projection theorem and projection operators, orthonormal sets, harmonic analysis on L2

Spectral theory: self-adjoint operators, positive operators, spectral families of self-adjoint operators, resolvent of a self-adjoint operator, eigenvalue problems for differential and integral equations.