Mappings with closed range and compactness

The motivation for this note is the result of E.O. Thorp that a normed linear space E is finite dimensional if and only if every continuous linear map for E into any normed linear space has a closed range. Here, a class of Hausdorff topological groups is introduced; called r-compactifiable topological groups, they include compact groups, locally compact Abelian groups and locally convex linear topological spaces. It is proved that a group in this class which is separable, complete metrizable or locally compact, is necessarily compact if its image by a continuous group homomorphism is necessarily closed. It is deduced then that a Hausdorff locally convex is zero if its image by a continuous additive map is necessarily closed. (author)