Syllabus
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1. Basic facts on operators Banach and Hilbert spaces. Operators and functionals, linear and nonlinear. Boundedness, continuity. Operator norm. Von Neumann series. *
2. An introduction to spectral analysis Eigenvalues, spectrum, resolvent set, spectral radius. *
3. Compact operators Compact operators, spectrum. *
4. Dual and adjoint operators Duality, dual operatost, dual spaces, representation theorems. Adjoint and self-adjoint operator, Hermite operator, their spectrum. Eigenfunction bases. *
5. Unbounded operators. Unbounded operator, closed operator. Differential operators. *
6. Special functions and polynomials Bases in Hilbert space, polynomial bases. Recurrent formulas for orthogonal polynomials. Special functions: Legendre, Laguerre, Hermite polynomials, hypergeometric series.
Annotation
An introduction to functional analysis, operator theory and special functions for physicists. It is the sequel to the basic 5-semester course of mathematics for physicists.