  It is shown that a Banach space $E$ has type $p$ if and only for some (all)
$d\ge 1$ the Besov space $B_{p,p}^{(\frac1p-\frac12)d}(\R^d;E)$ embeds into the
space $\g(L^2(\R^d),E)$ of $\g$-radonifying operators $L^2(\R^d)\to E$. A
similar result characterizing cotype $q$ is obtained. These results may be
viewed as $E$-valued extensions of the classical Sobolev embedding theorems.
