Injectivity and flatness of semitopological modules

The spaces D, S and E' over \mathbb{R}^(n) are known to be flat modules over A=\mathbb{C}[\partial_{1},...,\partial_{n}], whereas their duals D', S' and E are known to be injective modules over the same ring. Let A be a Noetherian k-algebra (k=\mathbb{R} or \mathbb{C}). The above observation leads us to study in this paper the link existing between the flatness of an A-module E which is a locally convex topological k-vector space and the injectivity of its dual. We show that, for dual pairs (E,E') which are (K) over A--a notion which is explained in the paper--, injectivity of E' is a stronger condition than flatness of E. A preprint of this paper (dated September 2009) has been quoted and discussed by Shankar.