Hilbert spaces

Abstract

Fourier series played a significant role in the development of Hilbert spaces and other aspects of abstract analysis. The theory of Hilbert spaces returns the favor by illuminating much of the information about Fourier series. We first develop enough information about Hilbert spaces to allow us to regard Fourier series as orthonormal expansions. We prove that (the symmetric partial sums of) the Fourier series of a square integrable function converges in $L^2$. From this basic result we obtain corollaries such as Parseval’s formula and the Riemann-Lebesgue lemma. We prove Bernstein’s theorem: the Fourier series of a Hölder continuous function (with exponent greater than ${1 \over 2}$) converges absolutely. We prove the spectral theorem for compact Hermitian operators. We include Sturm-Liouville theory to illustrate orthonormal expansion. We close by discussing spherical harmonics, indicating one way to pass from the circle to the sphere. These results leave one in awe at the strength of 19-th century mathematicians.