  General expository paper concerning topics in Hilbert spaces, spectral
theory, and harmonic analysis. The preliminary section includes basic Banach
algebra and Hilbert space theory with a digression on Riesz bases. The second
and third sections are focused on deriving the spectral theorem for Hilbert
spaces in finite and arbitrary dimension, respectively. The fourth and fifth
sections present applications of the general theory: first to finite abelian
groups, then to the space $L^2(S^1,\mu)$.
