In this article, we prove Herglotz's theorem for Hilbert-valued time series.
This requires the notion of an operator-valued measure, which we shall make
precise for our setting. Herglotz's theorem for functional time series allows
to generalize existing results that are central to frequency domain analysis on
the function space. In particular, we use this result to prove the existence of
a functional Cram{\'e}r representation of a large class of processes, including
those with jumps in the spectral distribution and long-memory processes. We
furthermore obtain an optimal finite dimensional reduction of the time series
under weaker assumptions than available in the literature. The results of this
paper therefore enable Fourier analysis for processes of which the spectral
density operator does not necessarily exist