  This paper presents a study of the inherent structural properties of Krylov
subspaces, in particular for the self-adjoint class of operators, and how they
relate with the important phenomenon of `Krylov solvability' of linear inverse
problems. Owing to the complexity of the problem in the unbounded setting,
recently developed perturbative techniques are used that exploit the use of the
weak topology on $\mathcal{H}$. We also make a strong connection between the
approximative properties of the Krylov subspace and the famous Hamburger
problem of moments, in particular the determinacy condition.
