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Dieudonné-Schwartz theorem on bounded sets in inductive limits


Authors: J. Kučera and K. McKennon
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
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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 [Enhancements On Off] (What's this?)

  • [1] J. Dieudonné and L. Schwartz, La dualité dans les espaces $ (\mathcal{F})$ et $ (\mathcal{L}\mathcal{F})$, Ann. Inst. Fourier (Grenoble) 1 (1949), 61-101. MR 0038553 (12:417d)
  • [2] J. Horváth, Topological vector spaces and distributions, Vol. 1, Addison-Wesley, Reading, Mass., 1966.
  • [3] J. Kucera and K. McKennon, Bounded sets in inductive limits, Proc. Amer. Math. Soc. 69 (1978), 62-64. MR 0463937 (57:3875)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0553378-X
Keywords: Locally convex space, inductive limit, bounded set
Article copyright: © Copyright 1980 American Mathematical Society

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