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# Functional Analysis

Select from the following:
Metric spaces. Examples. Convergence of sequences. Cauchy’s
condition for convergence. Complete metric space. Cantor’s 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. Parseval’s 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.

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