Separable Hilbert Spaces

Abstract

A basis of a Hilbert space \(\mathcal{H}\) is a set B of vectors such that the closed linear hull of B equals \(\mathcal{H}\). A Hilbert space is called separable if it has a countable basis. The Gram-Schmidt orthonormalization proves that every separable Hilbert space has an orthonormal basis. We give several characterizations of an orthonormal basis involving the Fourier expansion, the completness relation, and the Parseval relation. It follows that every infinite dimensional separable Hilbert space over the field \(\mathbb K\) is isomorphic to the sequence space \(\ell^2(\mathbb K)\). Next we investigate weight functions ρ and the related system of functions \(\rho_n(x)= x^n \rho(x)\) \(n=0,1,2,\ldots\) in the Hilbert space \(L^2(I,\,\textrm{d}x)\) \(I=[a,b]\) \(-\infty \leq a <b \leq \infty\), and give sufficient conditions under which the Gram-Schmidt orthonormalization of this system of functions is an orthonormal basis and prove some important properties of the resulting orthonormal polynomials, the “Knotensatz”. We conclude with some concrete examples of complete orthonormal systems for the Hilbert space \(L^2(I,\rho \,\textrm{d}x)\): for \(I=\mathbb{R}\) with \(\rho(x)=\,\textrm{e}^{-x^2}\) we get the Hermite polynomials, for \(I=[0,\infty)\) and \(\rho(x)=\,\textrm{e}^{-x}\) the Laguerre polynomials, and for \(I=[-1,1]\) and \(\rho(x)=1\) the Legendre polynomials.