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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Partitions of unity and a closed embedding theorem for $(C^{p},b^*)$-manifolds
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by Richard E. Heisey PDF
Trans. Amer. Math. Soc. 206 (1975), 281-294 Request permission

Abstract:

Many manifolds of fiber bundle sections possess a natural atlas $\{ ({U_\alpha },{\phi _\alpha })\}$ such that the transition maps ${\phi _\beta }\phi _\alpha ^{ - 1}$, in addition to being smooth, are continuous with respect to the bounded weak topology of the model. In this paper we formalize the idea of such manifolds by defining $({C^p},{b^\ast })$-manifolds, $({C^p},{b^\ast })$-morphisms, etc. We then show that these manifolds admit $({C^p},{b^\ast })$-partitions of unity subordinate to certain open covers and that they can be embedded as closed $({C^p},{b^\ast })$-submanifolds of their model. A corollary of our work is that for any Banach space $B$, the conjugate space ${B^\ast }$ admits smooth partitions of unity subordinate to covers by sets open in the bounded weak-$\ast$ topology.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 206 (1975), 281-294
  • MSC: Primary 58B05; Secondary 58C20
  • DOI: https://doi.org/10.1090/S0002-9947-1975-0397767-8
  • MathSciNet review: 0397767