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Abstract. For a (DF)-space E and a tensor norm α we investigate the derivatives Tor l α (E, ·) of the tensor product functor E ⊗ α · : F S → LS from the category of Fréchet spaces to the category of linear spaces. Necessary and sufficient conditions for the vanishing of Tor 1 α (E, F ), which is strongly related to the exactness of tensored sequences, are presented and characterizations in the nuclear and (co-)echelon cases are given.

In this article we characterize the quasi‐barrelledness of the projective tensor product of a coechelon space of type one k 1(A) with a Fréchet space, including homological conditions as exactness properties of the corresponding tensor product functor k 1(A) ·: ℱ → ℒ︁, acting from the category of Fréchet spaces to the category of linear spaces, resp. the vanishing of its first right derivative Tor1π (k 1(A),.). This generalizes and extends a classical theorem of A. Grothendieck ([13, Chap. II, §4, No. 3, Theorem 15]). Further we present an analogous theorem for complete coechelon spaces of type zero and the injective tensor product and results concerning the stronger property of barrelledness. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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We prove a decomposition lemma for elementary tensors and present an analogous result of a (DN)-(Ω) splitting theorem for the theory of tensoring an exact sequence of Fréchet spaces with a (DF)-space. This result is free of nuclearity and hilbertisability assumptions and can be applied in a natural way to vector-valued linear partial differential operators.