  We prove that the space of persistence diagrams on $n$ points (with the
bottleneck or a Wasserstein distance) coarsely embeds into Hilbert space by
showing it is of asymptotic dimension $2n$. Such an embedding enables
utilisation of Hilbert space techniques on the space of persistence diagrams.
We also prove that when the number of points is not bounded, the corresponding
spaces of persistence diagrams do not have finite asymptotic dimension.
Furthermore, in the case of the bottleneck distance, the corresponding space
does not coarsely embed into Hilbert space.
