Distributions and Their Hermite Expansions

We present a self-contained treatment of the technical parts of distribution theory needed in quantum field theory. The treatment is particularly suited for physicists since an absolute minimum of abstract functional analysis is used: In fact, only the Baire category theorem is needed. The simple nature of some proofs depends on extensive use of the expansion of a distribution as a sum of harmonic oscillator wave-functions. While this Hermite expansion is not new, the fact that it provides elementary proofs of several theorems does appear to be new.