  In this paper, we set up the theoretical foundations for a high-dimensional
functional factor model approach in the analysis of large cross-sections
(panels) of functional time series (FTS). We first establish a representation
result stating that, under mild assumptions on the covariance operator of the
cross-section, we can represent each FTS as the sum of a common component
driven by scalar factors loaded via functional loadings, and a mildly
cross-correlated idiosyncratic component. Our model and theory are developed in
a general Hilbert space setting that allows for mixed panels of functional and
scalar time series. We then turn to the identification of the number of
factors, and the estimation of the factors, their loadings, and the common
components. We provide a family of information criteria for identifying the
number of factors, and prove their consistency. We provide average error bounds
for the estimators of the factors, loadings, and common component; our results
encompass the scalar case, for which they reproduce and extend, under weaker
conditions, well-established similar results. Under slightly stronger
assumptions, we also provide uniform bounds for the estimators of factors,
loadings, and common component, thus extending existing scalar results. Our
consistency results in the asymptotic regime where the number $N$ of series and
the number $T$ of time observations diverge thus extend to the functional
context the "blessing of dimensionality" that explains the success of factor
models in the analysis of high-dimensional (scalar) time series. We provide
numerical illustrations that corroborate the convergence rates predicted by the
theory, and provide finer understanding of the interplay between $N$ and $T$
for estimation purposes. We conclude with an application to forecasting
mortality curves, where we demonstrate that our approach outperforms existing
methods.
