Algebraic characterisation of hyperspace corresponding to topological vector space

Document Type : Research Paper

Authors

1 Department of Mathematics, Vivekananda College,Thakurpukur, Kolkata, West Bengal, India

2 Department of Pure Mathematics, University of Calcutta, Kolkata, West Bengal, India

Abstract

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 evs
structure. 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.

Keywords

References

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Journal of Hyperstructures
Volume 11, Issue 1
June 2022
Pages 48-64

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