On the Schwartz space isomorphism theorem for the Riemannian symmetric spaces

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.