Acyclic inductive spectra of Fréchet spaces

Jochen Wengenroth

Studia Mathematica (1996)

  • Volume: 120, Issue: 3, page 247-258
  • ISSN: 0039-3223

Abstract

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We provide new characterizations of acyclic inductive spectra of Fréchet spaces which improve the classical theorem of Palamodov and Retakh. It turns out that acyclicity, sequential retractivity (defined by Floret) and further strong regularity conditions (introduced e.g. by Bierstedt and Meise) are all equivalent. This solves a problem that was folklore since around 1970. For inductive limits of Fréchet-Montel spaces we obtain even stronger results, in particular, Grothendieck's problem whether regular (LF)-spaces are complete has a positive solution in this case and we show that even the weakest regularity conditions already imply acyclicity. One of the main benefits from our results is an improvement in the theory of projective spectra of (DFM)-spaces. We prove the missing implication in a theorem of Vogt and thus obtain evaluable conditions for vanishing of the derived projective limit functor which have direct applications to classical problems of analysis like surjectivity of partial differential operators on various classes of ultradifferentiable functions (as was explained e.g. by Braun, Meise and Vogt).

How to cite

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Wengenroth, Jochen. "Acyclic inductive spectra of Fréchet spaces." Studia Mathematica 120.3 (1996): 247-258. <http://eudml.org/doc/216335>.

@article{Wengenroth1996,
abstract = {We provide new characterizations of acyclic inductive spectra of Fréchet spaces which improve the classical theorem of Palamodov and Retakh. It turns out that acyclicity, sequential retractivity (defined by Floret) and further strong regularity conditions (introduced e.g. by Bierstedt and Meise) are all equivalent. This solves a problem that was folklore since around 1970. For inductive limits of Fréchet-Montel spaces we obtain even stronger results, in particular, Grothendieck's problem whether regular (LF)-spaces are complete has a positive solution in this case and we show that even the weakest regularity conditions already imply acyclicity. One of the main benefits from our results is an improvement in the theory of projective spectra of (DFM)-spaces. We prove the missing implication in a theorem of Vogt and thus obtain evaluable conditions for vanishing of the derived projective limit functor which have direct applications to classical problems of analysis like surjectivity of partial differential operators on various classes of ultradifferentiable functions (as was explained e.g. by Braun, Meise and Vogt).},
author = {Wengenroth, Jochen},
journal = {Studia Mathematica},
keywords = {inductive and projective limits; acyclicity; derived projective limit functor; acyclic inductive spectra of Fréchet spaces; theorem of Palamodov and Retakh; inductive limits of Fréchet-Montel spaces; regular (LF)-spaces; projective spectra of (DFM)-spaces; projective limit functor; surjectivity of partial differential operators},
language = {eng},
number = {3},
pages = {247-258},
title = {Acyclic inductive spectra of Fréchet spaces},
url = {http://eudml.org/doc/216335},
volume = {120},
year = {1996},
}

TY - JOUR
AU - Wengenroth, Jochen
TI - Acyclic inductive spectra of Fréchet spaces
JO - Studia Mathematica
PY - 1996
VL - 120
IS - 3
SP - 247
EP - 258
AB - We provide new characterizations of acyclic inductive spectra of Fréchet spaces which improve the classical theorem of Palamodov and Retakh. It turns out that acyclicity, sequential retractivity (defined by Floret) and further strong regularity conditions (introduced e.g. by Bierstedt and Meise) are all equivalent. This solves a problem that was folklore since around 1970. For inductive limits of Fréchet-Montel spaces we obtain even stronger results, in particular, Grothendieck's problem whether regular (LF)-spaces are complete has a positive solution in this case and we show that even the weakest regularity conditions already imply acyclicity. One of the main benefits from our results is an improvement in the theory of projective spectra of (DFM)-spaces. We prove the missing implication in a theorem of Vogt and thus obtain evaluable conditions for vanishing of the derived projective limit functor which have direct applications to classical problems of analysis like surjectivity of partial differential operators on various classes of ultradifferentiable functions (as was explained e.g. by Braun, Meise and Vogt).
LA - eng
KW - inductive and projective limits; acyclicity; derived projective limit functor; acyclic inductive spectra of Fréchet spaces; theorem of Palamodov and Retakh; inductive limits of Fréchet-Montel spaces; regular (LF)-spaces; projective spectra of (DFM)-spaces; projective limit functor; surjectivity of partial differential operators
UR - http://eudml.org/doc/216335
ER -

References

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