A new characterization of 𝑃𝑟𝑜𝑗¹𝒳=0 for countable spectra of (LB)-spaces

Wengenroth, Jochen

doi:10.1090/S0002-9939-99-04559-1

A new characterization of $\operatorname {Proj}^1{\mathcal X}=0$ for countable spectra of (LB)-spaces
HTML articles powered by AMS MathViewer

by Jochen Wengenroth

Proc. Amer. Math. Soc. 127 (1999), 737-744

DOI: https://doi.org/10.1090/S0002-9939-99-04559-1

PDF | Request permission

Abstract:

The derived projective limit functor Proj$^1$ is a very useful tool for investigating surjectivity problems in various parts of analysis (e.g. solvability of partial differential equations). We provide a new characterization for vanishing Proj$^1$ on projective spectra of (LB)-spaces which improves a classical result of V. P. Palamodov and V. S. Retakh.

References

J. Bonet, P. Domański, Real analytic curves in Fréchet spaces and their duals, Monatsh. Math., to appear.
N. Bourbaki, Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris, 1971. MR 0358652
Rüdiger W. Braun, Reinhold Meise, and Dietmar Vogt, Applications of the projective limit functor to convolution and partial differential equations, Advances in the theory of Fréchet spaces (Istanbul, 1988) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 287, Kluwer Acad. Publ., Dordrecht, 1989, pp. 29–46. MR 1083556
R. W. Braun, R. Meise, and D. Vogt, Existence of fundamental solutions and surjectivity of convolution operators on classes of ultra-differentiable functions, Proc. London Math. Soc. (3) 61 (1990), no. 2, 344–370. MR 1063049, DOI 10.1112/plms/s3-61.2.344
Rüdiger W. Braun and Dietmar Vogt, A sufficient condition for $\textrm {Proj}^1{\scr X}=0$, Michigan Math. J. 44 (1997), no. 1, 149–156. MR 1439674, DOI 10.1307/mmj/1029005626
L. Frerick and J. Wengenroth, A sufficient condition for vanishing of the derived projective limit functor, Arch. Math. (Basel) 67 (1996), no. 4, 296–301. MR 1407332, DOI 10.1007/BF01197593
V. P. Palamodov, The projective limit functor in the category of topological linear spaces, Mat. Sb. (N.S.) 75 (117) (1968), 567–603 (Russian). MR 0223851
V. P. Palamodov, Homological methods in the theory of locally convex spaces, Uspehi Mat. Nauk 26 (1971), no. 1(157), 3–65 (Russian). MR 0293365
V. S. Retah, The subspaces of a countable inductive limit, Dokl. Akad. Nauk SSSR 194 (1970), 1277–1279 (Russian). MR 0284794
Dietmar Vogt, On the functors $\textrm {Ext}^1(E,F)$ for Fréchet spaces, Studia Math. 85 (1987), no. 2, 163–197. MR 887320, DOI 10.4064/sm-85-2-163-197
D. Vogt, Lectures on projective spectra of (DF)-spaces, Seminar lectures, AG Funktionalanalysis Düsseldorf/Wuppertal (1987).
Dietmar Vogt, Topics on projective spectra of (LB)-spaces, Advances in the theory of Fréchet spaces (Istanbul, 1988) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 287, Kluwer Acad. Publ., Dordrecht, 1989, pp. 11–27. MR 1083555
Dietmar Vogt, Regularity properties of (LF)-spaces, Progress in functional analysis (Peñíscola, 1990) North-Holland Math. Stud., vol. 170, North-Holland, Amsterdam, 1992, pp. 57–84. MR 1150738, DOI 10.1016/S0304-0208(08)70311-6
Jochen Wengenroth, Acyclic inductive spectra of Fréchet spaces, Studia Math. 120 (1996), no. 3, 247–258. MR 1410451, DOI 10.4064/sm-120-3-247-258