  In most classical holomorphic function spaces on the unit disk in which the
polynomials are dense, a function $f$ can be approximated in norm by its
dilates $f_r(z):=f(rz)~(r<1)$, in other words, $\lim_{r\to1^-}\|f_r-f\|=0$. We
construct a de Branges-Rovnyak space ${\mathcal H}(b)$ in which the polynomials
are dense, and a function $f\in{\mathcal H}(b)$ such that
$\lim_{r\to1^-}\|f_r\|_{{\mathcal H}(b)}=\infty$. The essential feature of our
construction lies in the fact that $b$ is an outer function.
