We prove that if X is an infinite-dimensional Banach space with $C^p$ smooth partitions of unity then X and X∖ K are $C^p$ diffeomorphic for every weakly compact set K ⊂ X.

Albeverio, Kondratiev, and Röckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space ${\Gamma }_X$ of any manifold $X$. The name comes from the fact that various elements of the geometry of ${\Gamma }_X$ are constructed via lifting of the corresponding elements of the geometry of $X$. In this note, we construct a general algebraic framework for lifted geometry which can be applied to various “infinite dimensional spaces” associated to $X$. In order to define a lifted...

We study the size of the sets of gradients of bump functions on the Hilbert space ${\ell }_2$, and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in ${\ell }_2$ can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space ${\ell }_2$ can be uniformly approximated by $C^1$ smooth Lipschitz functions $\psi$ so that the cones generated by the ranges of its derivatives ${\psi }^{\text{'}}\left({\ell }_2\right)$ have empty interior. This implies that there are $C^1$ smooth Lipschitz bumps...