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

The bootstrap in kernel regression for stationary ergodic data when both response and predictor are functions