  Let $E$ be a Banach space and $\X$ be the closed unit ball of the dual space
$E^*$. For a compact set $K$ in $E$, we prove that $K$ is polynomially convex
in $E$ if and only if there exist a unital commutative Banach algebra $A$ and a
continuous function $f:\X\to A$ such that (1) $A$ is generated by $f(\X)$, (2)
the character space of $A$ is homeomorphic to $K$, and (3) $K=\vsp(f)$ the
joint spectrum of $f$. In case $E=\c(X)$, where $X$ is a compact Hausdorff
space, we will see that $\X$ can be replaced by $X$.
