I take an extension from $L^2(\mathbb R)$ to $\mathscr S'(\mathbb R)$ of tempered distributions for a mapping of nonlinear distribution. I do not want to use seminorm, but the norm, therefore the extension to $\mathscr S'(\mathbb R)$. $\mathscr S'(\mathbb R)$ is defined as the continuous dual of the Schwartz space, also $\mathscr S'(\mathbb R) \subset \mathscr D'(\mathbb R)$.
Does the norm exist on $\mathscr S'(\mathbb R)$?