  We analyze the possibility of defining infinite-dimensional manifolds as
ringed spaces. More precisely, we consider three definitions of manifolds
modeled on locally convex spaces: in terms of charts and atlases, in terms of
ringed spaces, and in terms of functored spaces, as introduced by Douady in his
thesis. It is shown that for large classes of locally convex model spaces
(containing Fr\'echet spaces and duals of Fr\'echet-Schwartz spaces), the three
definitions are actually equivalent. The equivalence of the definition via
charts with the definition via ringed spaces is based on the fact that for the
classes of model spaces under consideration, smoothness of maps turns out to be
equivalent to their scalarwise smoothness (that is, the smoothness of their
composition with smooth real-valued functions).
