  We consider the double functional nonparametric regression model
$Y=r(X)+\epsilon$, where the response variable $Y$ is Hilbert space-valued and
the covariate $X$ takes values in a pseudometric space. The data satisfy an
ergodicity criterion which dates back to Laib and Louani (2010) and are
arranged in a triangular array. So our model also applies to samples obtained
from spatial processes, e.g., stationary random fields indexed by the regular
lattice $\mathbb{Z}^N$ for some $N\in\mathbb{N}_+$. We consider a kernel
estimator of the Nadaraya--Watson type for the regression operator $r$ and
study its limiting law which is a Gaussian operator on the Hilbert space.
Moreover, we investigate both a naive and a wild bootstrap procedure in the
double functional setting and demonstrate their asymptotic validity. This is
quite useful as building confidence sets based on an asymptotic Gaussian
distribution is often difficult.
