Hyperspaces of topological vector spaces: their embedding in topological vector spaces

Abstract

Let L L be a real (Hausdorff) topological vector space. The space K [ L ] \mathcal {K}[L] of nonempty compact subsets of L L forms a (Hausdorff) topological semivector space with singleton origin when K [ L ] \mathcal {K}[L] is given the uniform (equivalently, the finite) hyperspace topology determined by L L . Then K [ L ] \mathcal {K}[L] is locally compact iff L L is so. Furthermore, K Q [ L ] \mathcal {K}\mathcal {Q}[L] , the set of nonempty compact convex subsets of L L , is the largest pointwise convex subset of K [ L ] \mathcal {K}[L] and is a cancellative topological semivector space. For any nonempty compact and convex set X ⊂ L X \subset L , the collection K Q [ X ] ⊂ K Q [ L ] \mathcal {K}\mathcal {Q}[X] \subset \mathcal {K}\mathcal {Q}[L] is nonempty compact and convex. L L is iseomorphically embeddable in K Q [ L ] \mathcal {K}\mathcal {Q}[L] and, in turn, there is a smallest vector space L \mathcal {L} in which K Q [ L ] \mathcal {K}\mathcal {Q}[L] is algebraically embeddable (as a cone). Furthermore, when L L is locally convex, L \mathcal {L} can be given a locally convex vector topology I \mathcal {I} such that the algebraic embedding of K Q [ L ] \mathcal {K}\mathcal {Q}[L] in L \mathcal {L} is an iseomorphism, and then L \mathcal {L} is normable iff L L is so; indeed, I \mathcal {I} can be so chosen that, when L L is normed, the embedding of L L in K Q [ L ] \mathcal {K}\mathcal {Q}[L] and that of K Q [ L ] \mathcal {K}\mathcal {Q}[L] in L \mathcal {L} are both iseometries.