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

Abstract:

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