Spectral Theorem for Bounded Self-adjoint Operators

Diagonalization is one of the most important topics one learns in an elementary linear algebra course. Unfortunately, it only works on finite dimensional vector spaces, where linear operators can be represented by finite matrices. Later, one encounters infinite dimensional vector spaces (spaces of sequences, for example), where linear operators can be thought of as ”infinite matrices”. Extending the idea of diagonalization to these operators requires some new machinery. We present it below for the (relatively simple) case of bounded self-adjoint operators. It is important to note that this generalization is not merely a heuristic desire: infinite dimensions are inescapable. Indeed, mathematical physics is necessarily done in an infinite dimensional setting. Moreover, quantum theory requires the careful study of functions of operators on these spaces – the functional calculus. This may seem awfully abstract at first, but an example of a function of operators is known to anyone familiar with systems of linear ODEs. Given a system of ordinary linear differential equation of the form x′(t) = Ax(t)