2007
DOI: 10.4064/sm180-1-4
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On the derived tensor product functors for (DF)- and Fréchet spaces

Abstract: 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.

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Cited by 5 publications
(9 citation statements)
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“…We will need some homological definitions. For details and results on this subject we refer to [19] and [18].…”
Section: Applicationsmentioning
confidence: 99%
See 4 more Smart Citations
“…We will need some homological definitions. For details and results on this subject we refer to [19] and [18].…”
Section: Applicationsmentioning
confidence: 99%
“…Instead of recalling the definition of the derivatives we remark: If the above tensor product functor is left exact, for instance this is true if E is a nuclear (DF)-space or a coechelon space of type one, then Tor 1 π (E, F ) = 0 if and only if the tensor product functor is exact on all short exact sequences of Fréchet spaces starting with F (c.f. [19,Remark 2.2]). This easy remark shows the importance of the first derivative and its computation.…”
Section: Applicationsmentioning
confidence: 99%
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