Abstract:

A collection of sets in a normed vector space is said to be locally finite if every point has a neighborhood that meets finitely many of the sets. In every separable normed vector space, every collection that is locally finite in the norm topology can be expanded to a locally finite collection of sets that are actually open in the weak topology. A proof of this will be outlined and some results in the nonseparable case will be discussed.

On the one hand, the theorem extends to all Hilbert spaces (although the proof is considerably harder). On the other hand, it fails for l_{infinity}, and its status for inner product spaces of uncountable dimension (for example, the space of almost periodic functions) is an open problem.

A preprint is available at http://www.math.sc.edu/~nyikos/preprints.html