  We deduce a proof of the isomorphism theorem for certain closed subspace $\mc
S^p_\Gamma(X)$ of the $L^p$-Schwartz class functions $(0< p \leq 2) $ on a
Riemannian symmetric space $X$ where $\Gamma$ is a finite subset of
$\what{K}_M$. The Fourier transform considered is the Helgason Fourier
transform. Our proof relies only on the Paley-Wiener theorem for the
corresponding class of functions and hence it does not use the complicated
higher asymptotics of the elementary spherical functions.
