Dieudonné-Schwartz theorem in inductive limits of metrizable spaces

Qiu, Jing Hui

doi:10.1090/S0002-9939-1984-0754714-5

Dieudonné-Schwartz theorem in inductive limits of metrizable spaces
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by Jing Hui Qiu

Proc. Amer. Math. Soc. 92 (1984), 255-257

DOI: https://doi.org/10.1090/S0002-9939-1984-0754714-5

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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

Topological vector spaces and distributions

J. Kučera and K. McKennon, Bounded sets in inductive limits, Proc. Amer. Math. Soc. 69 (1978), no. 1, 62–64. MR 463937, DOI 10.1090/S0002-9939-1978-0463937-1
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, DOI 10.1090/S0002-9939-1982-0671201-1
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
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, DOI 10.1090/S0002-9939-1980-0553378-X