This book presents a thorough discussion of the theory of abstract inverse linear problems on Hilbert space. Given an unknown vectorfin a Hilbert spaceH, a linear operatorAacting onH, and a vectorginHsatisfyingAf=g, one is interested in approximatingfby finite linear combinations ofg,Ag,A2g,A3g, … The closed subspace generated by the latter vectors is called the Krylov subspace ofHgenerated bygandA. The possibility of solving this inverse problem by means of projection methods on the Krylov subspace is the main focus of this text.After giving a broad introduction to the subject, examples...