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Metric spaces. Examples. Convergence of sequences. Cauchyís condition for convergence. Complete metric space. Cantorís Intersection Theorem. Dense sets. Continuous mappings.
Function spaces. Examples.
Hilbert space. Infinite-dimensional vectors. Inner product, orthogonal functions, Pythagorean theorem
Expansion by orthogonal systems of functions in Hilbert space. Complete orthogonal systems. Orthonormal systems. Generalized Fourier series, coefficients. Parsevalís identity.
Concept of an operator. Examples of linear operators. Integral operator. Bounded, adjoint operators.
Integral equations. Kernel. Eigenvectors and eigenvalues. Self-adjoint linear operators.
Functional. Linear functional. Continuous linear functional. Conjugate space, dual space. Dual basis. Annihilator.