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

**[1]**Ernest Michael,*Topologies on spaces of subsets*, Trans. Amer. Math. Soc.**71**(1951), 152–182. MR**0042109**, 10.1090/S0002-9947-1951-0042109-4**[2]**Prem Prakash and Murat R. Sertel,*Topological semivector spaces: convexity and fixed point theory*, Semigroup Forum**9**(1974/75), no. 2, 117–138. MR**0374867****[3]**-,*On the continuity of Cartesian product and factorisation*, Discussion Paper no. 82, The Center for Mathematical Studies in Economics and Management Science, Northwestern University, Evanston, Ill., 1974. (Also issued as Preprint Series No. I/ 74-16, Easter 1974, International Institute of Management, D--1000 Berlin 33, Griegstrasse 5.)**[4]**Hans Rådström,*An embedding theorem for spaces of convex sets*, Proc. Amer. Math. Soc.**3**(1952), 165–169. MR**0045938**, 10.1090/S0002-9939-1952-0045938-2

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DOI:
https://doi.org/10.1090/S0002-9939-1976-0425881-3

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