  Let $X$ be a separable Banach space and $u{:} X\to\Bbb{R}$ locally upper
bounded. We show that there are a Banach space $Z$ and a holomorphic function
$h{:} X\to Z$ with $u(x)<\|h(x)\|$ for $x\in X$. As a consequence we find that
the sheaf cohomology group $H^q(X,\Cal{O})$ vanishes if $X$ has the bounded
approximation property (i.e., $X$ is a direct summand of a Banach space with a
Schauder basis), $\Cal{O}$ is the sheaf of germs of holomorphic functions on
$X$, and $q\ge1$. As another consequence we prove that if $f$ is a $C^1$-smooth
$\overline\partial$-closed $(0,1)$-form on the space $X=L_1[0,1]$ of summable
functions, then there is a $C^1$-smooth function $u$ on $X$ with
$\overline\partial u=f$ on $X$.
