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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

Authors: Prakash Prem and Murat R. Sertel
Journal: Proc. Amer. Math. Soc. 61 (1976), 163-168
MSC: Primary 54B20
MathSciNet review: 0425881
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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.

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Keywords: Topological vector space, topological semivector space, compact convex subsets, hyperspace, locally convex vector space, normable vector space, embedding, iseomorphism, cancellative topological semivector space
Article copyright: © Copyright 1976 American Mathematical Society