  A classical problem in general relativity is the Cauchy problem for the
linearised Einstein equation (the initial value problem for gravitational
waves) on a globally hyperbolic vacuum spacetime. A well-known result is that
it is uniquely solvable up to gauge solutions, given initial data on a
spacelike Cauchy hypersurface. The solution map is an isomorphism between
initial data (modulo gauge producing initial data) and solutions (modulo gauge
solutions). In the first part of this work, we show that the solution map is
actually an isomorphism of locally convex topological vector spaces. This
implies that the equivalence class of solutions depends continuously on the
equivalence class of initial data. We may therefore conclude well-posedness of
the Cauchy problem. In the second part, we show that the linearised constraint
equations can always be solved on a closed manifold with vanishing scalar
curvature. This generalises the classical notion of TT-tensors on flat space
used to produce models of gravitational waves. All our results are proven for
smooth and distributional initial data of arbitrary real Sobolev regularity.
