Hilbert spaces: Inner products, orthogonality, orthogonal projections, orthonormal bases, application to the space L2 and Fourier series.
Linear operators: nuclear operators, Hilbert-Schmidt operators, compact operators, bounded operators, unbounded operators. Basic properties.
Spectrum and singular values of a class of operators, inequalities between singular values, minimax properties, Schmidt development, elementary properties. Application to differential equations.
General theorems: Schauder’s theorem. Riesz theorem. Fredholm alternative. Allahverdiev’s theorem. Spectral decomposition theorem.