  In the first part we show that a vector-valued almost separably valued
function $f$ is holomorphic (harmonic) if and only if it is dominated by an
$L^1_\mathrm{loc}$ function and there exists a separating set $W\subset X'$
such that $\langle f,x'\rangle$ is holomorphic (harmonic) for all $x'\in W$.
This improves a known result which requires $f$ to be locally bounded. In the
second part we consider classical results in the $L^p$ theory for elliptic
differential operators of second order. In the vector-valued setting these
results are shown to be equivalent to the UMD property.
