In this paper we develop statistical inference tools for high dimensional functional time series. We introduce a new concept of physical dependent processes in the space of square integrable functions, which adopts the idea of basis decomposition of functional data in these spaces, and derive Gaussian and multiplier bootstrap approximations for sums of high dimensional functional time series. These results have numerous important statistical consequences. Exemplarily, we consider the

more »

... of joint simultaneous confidence bands for the mean functions and the construction of tests for the hypotheses that the mean functions in the spatial dimension are parallel. The results are illustrated by means of a small simulation study and in the analysis of Canadian temperature data.