Infinite-dimensional manifolds as ringed spaces

Abstract

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).