Splitting of exact sequences of Fréchet spaces in the absence of continuous norms

In the present paper we study the splitting of a short exact sequence 0 −→ G−→ E S −→ F −→ 0 of nuclear Fréchet spaces, where F need not admit a continuous norm, i.e., E or F may be spaces of continuous functions on an open subset of Rn or on a σ -compact C∞-manifold. If E is such a space, then we give a necessary and sufficient condition for the splitting of the sequence, in terms of a condition on S and of linear topological invariants (Ωloc) and (DNloc). This is used to give a structure theory of the space sN, a space which is isomorphic to many spaces of smooth functions, and to study the splitting of differential complexes. The paper offers a different approach to problems which have been studied in Doma´nski–Vogt [J. Funct. Anal. 153 (1998) 203].  2004 Elsevier Inc. All rights reserved.