A generalization of a theorem of A. Grothendieck

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 → ℒ, acting from the category of Fréchet spaces to the category of linear spaces, resp. the vanishing of its first right derivative Tor 1 π ( 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)