Given a closed manifold M and a closed regular submanifold L, consider the corresponding locally convex space \(I=I(M,L)\) of conormal distributions, with its natural topology, and the strong dual \(I'=I'(M,L)=I(M,L;\Omega )'\) of the space of conormal densities. It is shown that I is a barreled, ultrabornological, webbed, Montel, acyclic LF-space, and \(I'\) is a complete Montel space, which is a projective limit of bornological barreled spaces. In the case of codimension one, similar properties and additional descriptions are proved for the subspace \(K\subset I\) of conormal distributions supported in L and for its strong dual \(K'\) . We construct a locally convex Hausdoff space J and a continuous linear map \(I\rightarrow J\) such that the sequence \(0\rightarrow K\rightarrow I\rightarrow J\rightarrow 0\) as well as the transpose sequence \(0\rightarrow J'\rightarrow I'\rightarrow K'\rightarrow 0\) are short exact sequences in the category of continuous linear maps between locally convex spaces. Finally, it is shown that \(I\cap I'=C^\infty (M)\) in the space of distributions. In another publication, these results are applied to prove a Lefschetz trace formula for a simple foliated flow \(\phi =\{\phi ^t\}\) on a compact foliated manifold \((M,\mathcal {F})\) . It describes a Lefschetz distribution \(L_{\textrm{dis}}(\phi )\) defined by the induced action \(\phi ^*=\{\phi ^{t\,*}\}\) on the reduced cohomologies \({\bar{H}}^\bullet I(\mathcal {F})\) and \({\bar{H}}^\bullet I'(\mathcal {F})\) of the complexes of leafwise currents that are conormal and dual-conormal at the leaves preserved by \(\phi \) .
