The development of Krylov subspace methods for the solution of operator
equations has shown that two basic construction principles underlie the most
commonly used algorithms: the orthogonal residual (OR) and minimal residual
(MR) approaches. It is shown that these can both be formulated as
techniques for solving an approximation problem on a sequence of nested subspaces
of a Hilbert space, an abstract problem not necessarily related to an
operator equation. Essentially all Krylov subspace algorithms result when
these subspaces form a Krylov sequence. The well-known relations among
the iterates and residuals of MR/OR pairs are shown to hold also in this
rather general setting. We further show that a common error analysis for
these methods involving the canonical angles between subspaces allows many
of the known residual and error bounds to be derived in a simple and consistent
manner. An application of this analysis to compact perturbations of
the identity shows that MR/OR pairs of Krylov subspace methods converge
q-superlinearly when applied to such operator equations.