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.