  We develop a generalisation of Mercer's theorem to operator-valued kernels in
infinite dimensional Hilbert spaces. We then apply our result to deduce a
Karhunen-Lo\`eve theorem, valid for mean-square continuous Hilbertian
functional data, i.e. flows in Hilbert spaces. That is, we prove a series
expansion with uncorrelated coefficients for square-integrable random flows in
a Hilbert space, that holds uniformly over time.
