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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University of Mohaghegh Ardabili
2023-06-01
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| Series: | Journal of Hyperstructures |
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| Online Access: | https://jhs.uma.ac.ir/article_2527_28bef662e4b8fca59745563f67cc4f1c.pdf |
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| author | Jayeeta Saha Sandip Jana |
| author_facet | Jayeeta Saha Sandip Jana |
| author_sort | Jayeeta Saha |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-d4eef53a000f4d65af526c586ab0e5f1 |
| institution | Kabale University |
| issn | 2251-8436 2322-1666 |
| language | English |
| publishDate | 2023-06-01 |
| publisher | University of Mohaghegh Ardabili |
| record_format | Article |
| series | Journal of Hyperstructures |
| spelling | doaj-art-d4eef53a000f4d65af526c586ab0e5f12025-08-20T03:28:49ZengUniversity of Mohaghegh ArdabiliJournal of Hyperstructures2251-84362322-16662023-06-01111486410.22098/jhs.2023.25272527Algebraic characterisation of hyperspace corresponding to topological vector spaceJayeeta Saha0Sandip Jana1Department of Mathematics, Vivekananda College,Thakurpukur, Kolkata, West Bengal, IndiaDepartment of Pure Mathematics, University of Calcutta, Kolkata, West Bengal, IndiaLet 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.https://jhs.uma.ac.ir/article_2527_28bef662e4b8fca59745563f67cc4f1c.pdfexponential vector spacetopological exponential vector spacehyperspacesorder-isomorphismprimitive function |
| spellingShingle | Jayeeta Saha Sandip Jana Algebraic characterisation of hyperspace corresponding to topological vector space Journal of Hyperstructures exponential vector space topological exponential vector space hyperspaces order-isomorphism primitive function |
| title | Algebraic characterisation of hyperspace corresponding to topological vector space |
| title_full | Algebraic characterisation of hyperspace corresponding to topological vector space |
| title_fullStr | Algebraic characterisation of hyperspace corresponding to topological vector space |
| title_full_unstemmed | Algebraic characterisation of hyperspace corresponding to topological vector space |
| title_short | Algebraic characterisation of hyperspace corresponding to topological vector space |
| title_sort | algebraic characterisation of hyperspace corresponding to topological vector space |
| topic | exponential vector space topological exponential vector space hyperspaces order-isomorphism primitive function |
| url | https://jhs.uma.ac.ir/article_2527_28bef662e4b8fca59745563f67cc4f1c.pdf |
| work_keys_str_mv | AT jayeetasaha algebraiccharacterisationofhyperspacecorrespondingtotopologicalvectorspace AT sandipjana algebraiccharacterisationofhyperspacecorrespondingtotopologicalvectorspace |