  The discretization of least-squares problems for linear ill-posed operator
equations in Hilbert spaces is considered. The main subject of this article
concerns conditions for convergence of the associated discretized minimum-norm
least-squares solution to the exact solution using exact attainable data. The
two cases of global convergence (convergence for all exact solution) or local
convergence (convergence for a specific exact solution) are investigated. We
review the existing results and prove new equivalent condition when the
discretized solution always converges to the exact solution. An important tool
is to recognize the discrete solution operator as oblique projection. Hence,
global convergence can be characterized by certain subspaces having uniformly
bounded angles. We furthermore derive practically useful conditions when this
holds and put them into the context of known results. For local convergence we
generalize results on the characterization of weak or strong convergence and
state some new sufficient conditions. We furthermore provide an example of a
bounded sequence of discretized solutions which does not converge at all, not
even weakly.
