A generalization of a theorem of A. Grothendieck

Abstract

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)

On the derived tensor product functors for (DF)- and Fréchet spaces

“…3, Theorem 1] A. Grothendieck gave a comprehensive characterization of topological properties (to be more precise, of the quasi-barrelledness) of the (complete) projective tensor product of a (DF)-space with a Fréchet space in the case of Köthe coechelon spaces k 1 (A) and echelon spaces λ 1 (B). In [24] the author generalizes…”

“…In the same paper a corresponding theorem for the injective tensor product is also given. As the proof in [24] shows, we get: Theorem 5.10. Let E be either a coechelon space k 1 (A) of type one and α = π, or a complete coechelon space k 0 (A) of type zero and α = ε, and let F be a Fréchet space.…”

“…In [24] the author generalized and extended a classical and deep result of A. Grothendieck, who characterized topological properties of the projective tensor product of a coechelon space and an echelon space of type one ([5, II, §4, no. 3, Theorem 1]).…”

“…Theoretical applications of the Tor-theory established here such as the characterization of quojections or the property (Ω) of D. Vogt and M. J. Wagner can be found in [24], while practical applications such as the solutions of (DF)-valued partial differential operators with constant coefficients will be treated in [25], where we shall use a method of H. J. Petzsche ([22]) for decomposing elementary tensors.…”

A decomposition lemma for elementary tensors

“…The starting point of our considerations is the following problem, which inspired different authors to further studies, compare [5], [7], [8], [18], [19], [24]: Let E be a complete (DF)-and F a Fréchet space. If 0 → F → G q → H → 0 is an exact sequence of Fréchet spaces the question arises whether the tensored sequence 0 → E ⊗ π F −→ E ⊗ π G id⊗q −→ E ⊗ π H → 0 is again exact, where π denotes the projective tensor product.…”