Approximate identities for the Schwartz space

We establish a sufficient and a necessary conditions for the convergence, in the Schwartz space topology, of the sequence obtained by the convolution of an arbitrary given approximate identity for $$L(\mathbb {R}^k)$$ , whose terms belongs to the Schwartz space $$\mathscr {S}(\mathbb {R}^k)$$ , with any arbitrary Schwartz function to this function. Additionally, we give an example of a sequence, $$(\psi _n)_{n=1}^{+\infty }$$ in $$\mathscr {S}(\mathbb {R}^k)$$ , that is an approximate identity for $$L(\mathbb {R}^k)$$ but for which there exists a function $$f\in \mathscr {S} (\mathbb {R}^k)$$ such that $$\psi _n *f$$ does not converge to f as $$n \rightarrow +\infty $$ with respect to the topology of $$\mathscr {S}(\mathbb {R}^k)$$ .