Every separable normed vector space has the property that every locally finite family of subsets expands to a locally finite family of open sets. This is true whether the topology used is the one induced by the norm or some coarser topology such as the weak topology. Remarkably enough, however, there is enough “elbow room” for the following to be true:
Theorem. Let $X$ be a separable normed vector space and let ${S_n : n in omega}$ be a family of subsets that is locally finite in the norm. Then there is a family of subsets $U_n supset S_n$ that are open in the weak topology and are locally finite in the norm topology.
What makes this unusual is that the weakly open sets are far bigger than the norm-open ones, being of finite codimension in the whole space. This talk will give an overview of related results, some of which are even stronger, and some contrasting results for nonseparable spaces such as $ell_infty$ (to which the theorem does not extend) and nonseparable Hilbert spaces (to which it does extend).
How to participate in this seminar:
1. Book your nearest ACE facility;
2. Notify the ACE contact person at the host institution (Darren Condon) to notify you will be participating.
No access to an ACE facility? Contact Maaike Wienk to arrange a temporary Visimeet licence for remote access (limited number of licences available – first come first serve)