DieudonnéSchwartz theorem in inductive limits of metrizable spaces. II
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 Proc. Amer. Math. Soc. 108 (1990), 171175 Request permission
Abstract:
The DieudonnéSchwartz Theorem for bounded sets in strict inductive limits does not hold for general inductive limits $E = {\text {ind lim }}{E_n}$ . It does if all the ${E_n}$ are Fréchet spaces and for any $n \in N$ there is $m\left ( n \right ) \in N$ such that $\bar E_n^{{E_p}} \subset {E_{m\left ( n \right )}}$ for all $p \geq m\left ( n \right )$. 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 and bounded in some $\left ( {{E_n},{\xi _n}} \right )$. Also, some interesting results for bounded sets in inductive limits of Fréchet spaces are given.References

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Additional Information
 © Copyright 1990 American Mathematical Society
 Journal: Proc. Amer. Math. Soc. 108 (1990), 171175
 MSC: Primary 46A05
 DOI: https://doi.org/10.1090/S00029939199009947791
 MathSciNet review: 994779