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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Bounded sets in inductive limits
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by J. Kučera and K. McKennon PDF
Proc. Amer. Math. Soc. 69 (1978), 62-64 Request permission


The Dieudonné-Schwartz theorem for bounded sets in strict inductive limits does not hold for general inductive limits. A set B bounded in an inductive limit $E = {\operatorname {ind}}\;\lim {E_n}$ of locally convex spaces may not be contained in any ${E_n}$. If, however, each ${E_n}$ is closed in E, then B is contained in some ${E_n}$, but may not be bounded there.
  • 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
  • John Horváth, Topological vector spaces and distributions. Vol. I, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0205028
  • J. L. Kelley and Isaac Namioka, Linear topological spaces, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. MR 0166578, DOI 10.1007/978-3-662-41914-4
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 69 (1978), 62-64
  • MSC: Primary 46M10; Secondary 46A05
  • DOI:
  • MathSciNet review: 0463937