Algebraic characterisation of hyperspace corresponding to topological vector space

Let X be a Hausdorff topological vector space over the field of real or complex numbers. When Vietoris topology is given, the hyperspace \com {\mathcal 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 \com {\mathcal X}{}, there are other hyperspaces namely \set{\mathcal X}{}, \set{\mathcal X}{Bal} \set{\mathcal X}{CV}, \set{\mathcal X}{N_\theta}, \set{\X}{S}, \set{\mathcal X}{\theta} which have the same structure. To characterise the hyperspaces \set{\mathcal X}{}, \com {\mathcal 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 \com {\mathcal X}{} as well as \set{\mathcal X}{} as exponential vector spaces.

Keywords:

Exponential vector space , topological exponential vector space , hyperspaces , order-isomorphism , primitive function