An Introduction to Functional Analysis

It is one of the major results of finite-dimensional linear algebra that all the eigenvalues of real symmetric matrices are real and that the eigenvectors of distinct eigenvalues are orthogonal. In this chapter we prove similar results for compact self-adjoint operators on infinite-dimensional HIlbert space: we show that the spectrum consists entirely of real eigenvalues (except perhaps zero), that the multiplicity of every non-zero eigenvalue is finite, and that the eigenvectors form an orthonormal basis for H. The last fact follows from the Hilbert-Schmidt Theorem, which allows us to write such operators in terms of their eigenvalues and eigenvectors.