On dense subspaces in a class of Fr\'echet function spaces on R^n

When dealing with concrete problems in a function space on R^n, it is sometimes helpful to have a dense subspace consisting of functions of a particular type, adapted to the problem under consideration. We give a theorem that allows one to write down many of such subspaces in commonly occurring Fr'echet function spaces. These subspaces are all of the form ${pf_0 | p\in{\cal P}}$where$f_0$isafixedfunctionand${\cal P}$ is an algebra of functions. Classical results like the Stone-Weierstrass theorem for polynomials and the completeness of the Hermite functions are related by this theorem.