Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Dieudonné-Schwartz theorem in inductive limits of metrizable spaces


Author: Jing Hui Qiu
Journal: Proc. Amer. Math. Soc. 92 (1984), 255-257
MSC: Primary 46A05
DOI: https://doi.org/10.1090/S0002-9939-1984-0754714-5
MathSciNet review: 754714
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Dieudonné-Schwartz Theorem for bounded sets in strict inductive limits does not hold for general inductive limits $ E = {\operatorname{ind}}\lim {{\text{E}}_{\text{n}}}$. It does if each $ \bar E_n^E \subset {E_{m\left( n \right)}}$ and all the $ {E_n}$ are Fréchet spaces. A counterexample shows that this condition is not necessary. When $ E$ is a strict inductive limit of metrizable spaces $ {E_n}$, this condition is equivalent to the condition that each bounded set in $ E$ is contained in some $ {E_n}$.


References [Enhancements On Off] (What's this?)

  • [1] J. Horváth, Topological vector spaces and distributions, Vol. 1, Addision-Wesley, Reading, Mass., 1966.
  • [2] J. Kucera and K. McKennon, Bounded sets in inductive limits, Proc. Amer. Math. Soc. 69 (1978), 62-64. MR 0463937 (57:3875)
  • [3] J. Kucera and C. Bosch, Dieudonné-Schwartz theorem on bounded sets in inductive limits. II, Proc. Amer. Math. Soc. 86 (1982), 392-394. MR 671201 (84b:46005)
  • [4] A. P. Robertson and W. J. Robertson, Topological vector spaces, Cambridge Univ. Press, 1964. MR 0162118 (28:5318)
  • [5] J. Kucera and K. McKennon, Dieudonné-Schwartz theorem on bounded sets in inductive limits, Proc. Amer. Math. Soc. 78 (1980), 366-368. MR 553378 (81d:46007)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46A05

Retrieve articles in all journals with MSC: 46A05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0754714-5
Keywords: Locally convex spaces, (strict) inductive limit, bounded set
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society