On the space of subsets of a uniform space

Abstract

The set g of closed nonempty subsets of a metric space E can be made into a metric space by means of the well-known Hausdorff metric (Hausdorff [3, ?28]). When E is complete, so is g (see e.g. Kuratowski [6, ?29, no. IV]; Price [7]; Bourbaki [1, Chapter IX, ?2, ex. 71). In a similar way, if E is a uniform space, the uniform structure on E induces on g a separated uniform structure, which we shall call the Hausdorff uniform structure on g, following J. L. Kelley [5]. In this case, it is no longer true that the completeness of E implies that of g (although, if E is compact, g is then complete; see e.g. Bourbaki [1, Chapter II, ?4, ex. 6]). An example of a complete uniform space E for which g is not complete is provided by taking for E any complete but not fully complete locally convex topological vector space, for the completeness of g implies the full completeness of E (see [5 ]). The first result of this paper shows that the subset of g consisting of all the (nonempty) compact sets in E is always complete, provided that E is complete. The second theorem gives an extension of this result, which, applied to a complete locally convex topological vector space, shows that the weakly compact sets also form a complete space. The third theorem is concerned with function spaces. The corresponding result for a space of linear mappings can be regarded as a generalization of the fact, that, for Banach spaces, the limit of a convergent sequence of weakly compact linear mappings is weakly compact. This is because convergence in norm of a sequence of linear mappings is precisely convergence, in the Hausdorff uniform structure on the range space, of the sequence of closed images of the unit ball. We have pleasure in thanking Dr. E. Michael for suggesting the generalization to uniform spaces of our results, originally stated for locally convex spaces, and for many helpful comments towards its achievement. Let E be a separated uniform space and ql a base for the uniform structure on E. If UCcl and A CE, we write U(A) for the set of points y with (x, y) E U for some xEA. On g, the set of closed nonempty subsets of E, the sets