Finite seminorm bound characterizes tempered distributions

Suppose $u$ is continuous. There is a basic zero-neighborhood $U=\{\phi :p_{{\alpha }_j{\beta }_j}(\phi )<{\epsilon }_j,1\le j\le r\}$ on which $\mathrm{\mid }u(\phi )\mathrm{\mid }<1$. If $r=0$, then $U=\mathcal{S}$ and linearity forces $u=0$, so take $C=0$. Otherwise put $Q(\phi )={\max }_j{p}_{{\alpha }_j{\beta }_j}(\phi )\mathrm{/}{\epsilon }_j$.

If $Q(\phi )>0$, then $\phi \mathrm{/}(2Q(\phi ))\in U$, whence $\mathrm{\mid }u(\phi )\mathrm{\mid }<2Q(\phi )$. If $Q(\phi )=0$, every scalar multiple of $\phi$ lies in $U$; boundedness of those scalar multiples of $u(\phi )$ forces $u(\phi )=0$. Choose $N$ and $M$ dominating the finitely many $\mathrm{\mid }{\alpha }_j\mathrm{\mid }$ and $\mathrm{\mid }{\beta }_j\mathrm{\mid }$. Then $Q(\phi )$ is at most ${\max }_j{\epsilon }_j^{-1}$ times the rectangular maximum in the statement, which proves the required estimate.

Conversely, assume the displayed estimate. For every $\epsilon >0$, the set on which its finite maximum is less than $\epsilon \mathrm{/}(C+1)$ is a zero-neighborhood and is carried by $u$ into the disk of radius $\epsilon$. Thus $u$ is continuous and hence tempered.