  We introduce polynomial processes taking values in an arbitrary Banach space
$B$ via their infinitesimal generator $L$ and the associated martingale
problem. We obtain two representations of the (conditional) moments in terms of
solutions of a system of ODEs on the truncated tensor algebra of dual
respectively bidual spaces. We illustrate how the well-known moment formulas
for finite dimensional or probability-measure valued polynomial processes can
be deduced in this general framework. As an application we consider polynomial
forward variance curve models which appear in particular as Markovian lifts of
(rough) Bergomi-type volatility models. Moreover, we show that the signature
process of a $d$-dimensional Brownian motion is polynomial and derive its
expected value via the polynomial approach.
