In this paper, we discuss the embeddability of subspaces of the
Gromov-Hausdorff space, which consists of isometry classes of compact metric
spaces endowed with the Gromov-Hausdorff distance, into Hilbert spaces. These
embeddings are particularly valuable for applications to topological data
analysis. We prove that its subspace consisting of metric spaces with at most n
points has asymptotic dimension n(n−1)/2. Thus, there exists a coarse
embedding of that space into a Hilbert space. On the contrary, if the number of
points is not bounded, then the subspace cannot be coarsely embedded into any
uniformly convex Banach space and so, in particular, into any Hilbert space.
Furthermore, we prove that, even if we restrict to finite metric spaces whose
diameter is bounded by some constant, the subspace still cannot be bi-Lipschitz
embedded into any finite-dimensional Hilbert space. We obtain both
non-embeddability results by finding obstructions to coarse and bi-Lipschitz
embeddings in families of isometry classes of finite subsets of the real line
endowed with the Euclidean-Hausdorff distance