A theorem on the Schwartz space of a reductive Lie group.

The purpose of this paper is to define the Fourier transform of an arbitrary tempered distribution on a reductive Lie group. To this end we define a topological vector space, [unk](G), in terms of the classes of irreducible unitary representations of G, which plays role of a dual Schwartz space. Our main theorem then asserts that the usual L(2) Fourier transform, when restricted to functions in the Schwartz space, [unk](G) defined by Harish-Chandra, provides a topological isomorphism from [unk](G) onto [unk](G).