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

MathSciNet review:
754714

Full-text PDF Free Access

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 . 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. Kučera and K. McKennon,*Bounded sets in inductive limits*, Proc. Amer. Math. Soc.**69**(1978), no. 1, 62–64. MR**0463937**, 10.1090/S0002-9939-1978-0463937-1**[3]**J. Kučera and C. Bosch,*Dieudonné-Schwartz theorem on bounded sets in inductive limits. II*, Proc. Amer. Math. Soc.**86**(1982), no. 3, 392–394. MR**671201**, 10.1090/S0002-9939-1982-0671201-1**[4]**A. P. Robertson and W. J. Robertson,*Topological vector spaces*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 53, Cambridge University Press, New York, 1964. MR**0162118****[5]**J. Kučera and K. McKennon,*Dieudonné-Schwartz theorem on bounded sets in inductive limits*, Proc. Amer. Math. Soc.**78**(1980), no. 3, 366–368. MR**553378**, 10.1090/S0002-9939-1980-0553378-X

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