Partitions of unity and a closed embedding theorem for -manifolds

Author:
Richard E. Heisey

Journal:
Trans. Amer. Math. Soc. **206** (1975), 281-294

MSC:
Primary 58B05; Secondary 58C20

MathSciNet review:
0397767

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Many manifolds of fiber bundle sections possess a natural atlas such that the transition maps , 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 -manifolds, -morphisms, etc. We then show that these manifolds admit -partitions of unity subordinate to certain open covers and that they can be embedded as closed -submanifolds of their model. A corollary of our work is that for any Banach space , the conjugate space admits smooth partitions of unity subordinate to covers by sets open in the bounded weak- topology.

**[1]**Robert Bonic and John Frampton,*Smooth functions on Banach manifolds*, J. Math. Mech.**15**(1966), 877–898. MR**0198492****[2]**J. Dieudonné,*Natural homomorphisms in Banach spaces*, Proc. Amer. Math. Soc.**1**(1950), 54–59. MR**0033973**, 10.1090/S0002-9939-1950-0033973-8**[3]**Nelson Dunford and Jacob T. Schwartz,*Linear Operators. I. General Theory*, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR**0117523****[4]**James Eells Jr.,*A setting for global analysis*, Bull. Amer. Math. Soc.**72**(1966), 751–807. MR**0203742**, 10.1090/S0002-9904-1966-11558-6**[5]**R. A. Graff,*Elements of local non-linear functional analysis*, Thesis, Princeton University, 1972.**[6]**David W. Henderson,*Stable classification of infinite-dimensional manifolds by homotopy-type*, Invent. Math.**12**(1971), 48–56. MR**0290413****[7]**Nicolaas H. Kuiper and Besseline Terpstra-Keppler,*Differentiable closed embeddings of Banach manifolds*, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 118–125. MR**0264709****[8]**J. Kurzweil,*On approximation in real Banach spaces*, Studia Math.**14**(1954), 214–231 (1955). MR**0068732****[9]**Kiiti Morita,*Star-finite coverings and the star-finite property*, Math. Japonicae**1**(1948), 60–68. MR**0026803****[10]**Richard S. Palais,*Banach manifolds of fiber bundle sections*, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 243–249. MR**0448405****[11]**Walter Rudin,*Real and complex analysis*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210528**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58B05,
58C20

Retrieve articles in all journals with MSC: 58B05, 58C20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0397767-8

Keywords:
Manifolds of sections,
bounded weak topology,
partitions of unity,
closed embedding

Article copyright:
© Copyright 1975
American Mathematical Society