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

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Abstract: The Dieudonné-Schwartz Theorem for bounded sets in strict inductive limits does not hold for general inductive limits . It does if each and all the are Fréchet spaces. A counterexample shows that this condition is not necessary. When is a strict inductive limit of metrizable spaces , this condition is equivalent to the condition that each bounded set in is contained in some .

**[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)**

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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