Dieudonné-Schwartz theorem on bounded sets in inductive limits
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- by J. Kučera and K. McKennon PDF
- Proc. Amer. Math. Soc. 78 (1980), 366-368 Request permission
Abstract:
The Dieudonné-Schwartz Theorem for bounded sets in strict inductive limits does not hold for general inductive limits $E = {\text {ind}}\;\lim {E_n}$. It does if every closed convex set in ${E_n}$ is closed in ${E_{n + 1}}$. This condition is not necessary. In case all spaces ${E_n}$ are normed a necessary and sufficient condition for the validity of the Dieudonné-Schwartz Theorem is given.References
- Jean Dieudonné and Laurent Schwartz, La dualité dans les espaces $\scr F$ et $(\scr L\scr F)$, Ann. Inst. Fourier (Grenoble) 1 (1949), 61–101 (1950) (French). MR 38553 J. Horváth, Topological vector spaces and distributions, Vol. 1, Addison-Wesley, Reading, Mass., 1966.
- 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
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 78 (1980), 366-368
- MSC: Primary 46A12; Secondary 46A09
- DOI: https://doi.org/10.1090/S0002-9939-1980-0553378-X
- MathSciNet review: 553378