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

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.