46Hxx Topological algebras, normed rings and algebras, Banach algebras (For group algebras, convolution algebras and measure algebras, see 43A10, 43A20)

46Jxx Commutative Banach algebras and commutative topological algebras [See also 46E25]

46Kxx Topological (rings and) algebras with an involution [See also 16W10]

46Lxx Selfadjoint operator algebras (C*-algebras, von Neumann (W*-) algebras, etc.) [See also 22D25, 47Lxx]

46Mxx Methods of category theory in functional analysis [See also 18-XX]

46Nxx Miscellaneous applications of functional analysis [See also 47Nxx]

46Sxx Other (nonclassical) types of functional analysis [See also 47Sxx]

46Txx Nonlinear functional analysis [See also 47Hxx, 47Jxx, 58Cxx, 58Dxx]

The aim of this course is to give a very modest introduction to the extremely rich and well-developed theory of Hilbert spaces, an introduction that hopefully will provide the students with a knowledge of some of the fundamental results of the theory and will make them familiar with everything needed in order to understand, believe and apply the spectral theorem for selfadjoint operators in Hilbert space. This implies that the course will have to give answers to such questions as - What is a Hilbert space? - What is a bounded operator in Hilbert space? - What is a selfadjoint operator in Hilbert space? - What is the spectrum of such an operator? - What is meant by a spectral decomposition of such an operator?