  We generalize the Gleason-Kahane-\.Zelazko theorem to modules. As an
application, we show that every linear functional on a Hardy space that is
non-zero on outer functions is a multiple of a point evaluation. A further
consequence is that every linear endomorphism of a Hardy space that maps outer
functions to nowhere-zero functions is a weighted composition operator. In
neither case is continuity assumed. We also consider some extensions to other
function spaces, including the Bergman, Dirichlet and Besov spaces, the little
Bloch space and VMOA.
