Many of the above-mentioned authors consider that ϵ = (ϵ1, …, ϵn)′ is an homoskedastic independent error vector, i.e. E(ϵ)=0, Var(ϵ)=σ2 and Cov(ϵi, ϵj) = 0, i ≠ j. This assumption is made to obtain simple diagnostics or confidence intervals for the response but it could be too restrictive in functional regression models and difficult to check or fulfill in practice. Some papers consider dependence in the functional variate. See, for example, [31, 32] and  for contributions devoted to spatial dependence with functional data or [34–36] and  for time dependence. In both cases, the functional nature of the variate complicates the predictive ability of the model. The aim of this paper is to extend the GLS approach  to the functional context as the simplest way of incorporating temporal or spatial dependence in the regression models. In fact, the GLS approach can handle a wide range of regression models with dependence in a simple way: equi-correlation models, random effects, time and spatial dependence, and so on. This idea was first introduced in the context of FDA in .