On holomorphic domination, I

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