  This monograph is centred at the intersection of three mathematical topics,
that are theoretical in nature, yet with motivations and relevance deep rooted
in applications: the linear inverse problems on abstract, in general
infinite-dimensional Hilbert space; the notion of Krylov subspace associated to
an inverse problem, i.e., the cyclic subspace built upon the datum of the
inverse problem by repeated application of the linear operator; the possibility
to solve the inverse problem by means of Krylov subspace methods, namely
projection methods where the finite-dimensional truncation is made with respect
to the Krylov subspace and the approximants converge to an exact solution to
the inverse problem.
