Lecture Notes on the Spectral Theorem

Sections 1 through 5 of these notes are from a series of lectures I gave in the summer of 1989. The object of these lectures was to give a reasonably self-contained proof of the Spectral Theorem for bounded normal operators on an infinite dimensional complex Hilbert space. They are aimed at second year graduate students who have at least had a bit of functional analysis. Since the talks, and in particular these notes, were meant to be informal, I did not take the time to carefully footnote the sources for the arguments which I stole from the standard texts. On can safely assume that the clever bits may be found—word for word—in either [5, Chapters 10–13] and/or [1, §1.1]. I added Section 6 since I was curious about non normal operators and what can be said about them. Section 7 was added in 2006 based on a series of talks in our functional analysis seminar. Section 6 comes from [5, §§10.21– 33]. Section 7 is based on some old lecture notes from a course I took from my advisor, Marc Rieffel, in 1978 and [5, Chap. 13].