ABSTRACT
In V→N, the spectral theorem for selfadjoint operators is summarised in Theorem. A selfadjoint operator in an infinite-dimensional vector space V→ ∞ may be expected to have a countably infinite number of eigenvalues with a corresponding countably infinite set of eigenvectors and eigenprojectors. One may also expect a similar spectral theorem to apply. This chapter introduces spectral functions for selfadjoint operators in V→N in a way which can be generalised to V→ ∞ A selfadjoint operator has associated with it a complete orthogonal family of eigenprojectors. These eigenprojectors determine a spectral function. Conversely every spectral function defines a selfadjoint operator. The one-to-one relation between spectral functions and selfadjoint operators is summarised in the spectral theorem. The importance of selfadjoint operators together with their spectral functions and spectral measures lie in their relation with probability distribution functions and probability measures.
