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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 thedoi:10.17877/de290r-20939 fatcat:gg2ee735jfajbaq2itefgqu2s4