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 weierp;(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 weierp;(X), there are other hyperspaces namely P(X), PBal(X) PCV (X), PN theta; (X), PS(X), P theta;(X) which have the same structure. To characterise the hyperspaces P(X), weierp;(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 weierp;(X) as well as P(X) as exponential vector spaces.