Theoretical Statistics and Mathematics Unit, ISI Delhi
Given a multi-valued mapping T between two structures A and B, it is always of interest to explore conditions that would guarantee for each x in A an appropriate choice of the values of T that it takes in T(x), so that the resulting well defined mapping enjoys certain regularity properties. One of the early and well known results belonging to this circle of ideas is the so called Michael’s selection theorem which guarantees the existence of a continuous selection of a (lower semicontinuous) set-valued map acting on a metric space and taking values which are nonempty, closed and convex subsets of a Banach space.
The present talk deals with this issue in the context of the well known Hahn Banach theorem involving (a) the extendability of a bounded linear functional from a subspace of a Banach space X to a bounded linear functional on the whole space (b) The separation of points of X by means of a bounded linear functional on X. In either case, the non-uniqueness of the resulting functional gives rise to a multi-valued correspondence between appropriate objects arising in the situation. Our main purpose shall be to address these questions in the setting of a non-linear analogue of the Hahn Banach theorem which is provided by an old result of McShane, stating that Lipschitz maps defined on a subset of a metric space may be extended- not necessarily uniquely- to a Lipschitz map on the whole space. Precisely, we shall deal with the questions involving the existence of a selection of the multi-valued maps arising from (a) and (b) above which is required to be a continuous linear map in the case of (a) and a bilipschitz map in the case of (b). The interplay between the existence of such selection maps and the geometry of the underlying Banach space will be the main focus of the discussion.