  In this paper we study $A$-projections, i.e. operators of a Hilbert space
$\HH$ which act as projections when a seminorm is considered in $\HH$.
$A$-projections were introduced by Mitra and Rao \cite{[MitRao74]} for finite
dimensional spaces. We relate this concept to the theory of compatibility
between positive operators and closed subspaces of $\HH$. We also study the
relationship between weighted least squares problems and compatibility.
