Algebraic characterisation of hyperspace corresponding to topological vector space
Let X be a Hausdor topological vector space over the field of real or complex numbers. When Vietoris topology is given,the hyperspace ℘(X) of all nonempty compact subsets of X forms a topological exponential vector space over the same field. Exponential vector space [shortly, evs] is an algebraic o...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Mohaghegh Ardabili
2023-06-01
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| Series: | Journal of Hyperstructures |
| Subjects: | |
| Online Access: | https://jhs.uma.ac.ir/article_2527_28bef662e4b8fca59745563f67cc4f1c.pdf |
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| Summary: | Let X be a Hausdor topological vector space over the field of real or complex numbers. When Vietoris topology is given,the hyperspace ℘(X) of all nonempty compact subsets of X forms a topological exponential vector space over the same field. Exponential vector space [shortly, evs] is an algebraic ordered extension of vector space in the sense that every evs contains a vector space, and conversely, every vector space can be embedded into such a structure. A semigroup structure, a scalar multiplication and a partial order with some compatible topology comprise the topological evsstructure. In this study, we have shown that besides ℘(X), there are other hyperspaces namely P(X), PBal(X) PCV (X), PNθ (X), PS(X), Pθ(X) which have the same structure. To characterise the hyperspaces P(X), ℘(X) in light of evs, we have introduced some properties of evs which remain invariant under order-isomorphism. We have also introduced the concept of primitive function of an evs, which plays an important role in such characterisation. Lastly, with the help of these properties, we have characterised ℘(X) as well as P(X) as exponential vector spaces. |
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| ISSN: | 2251-8436 2322-1666 |