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Transactions of the American Mathematical Society

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

 
 

 

Partitions of unity and a closed embedding theorem for $(C^{p},b^*)$-manifolds


Author: Richard E. Heisey
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
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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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Keywords: Manifolds of sections, bounded weak topology, partitions of unity, closed embedding
Article copyright: © Copyright 1975 American Mathematical Society