Hyperspaces and Superspaces: Linearity and Boundedness

Abstract

Continuing the previous research, we analyze constructions of hyperspaces and superspaces elaborating classifications of such spaces. The main emphasis is on linear hyperspaces, which are formed over linear spaces with additional structures such as norms and seminorms. Linear hyperspaces are constructed over multiseminormed linear spaces and their properties are studied. In particular, it is demonstrated that a linear hyperspace over a multiseminormed linear space is a multihyperseminormed linear space. This research is connected with problems of contemporary quantum physics solving some of these problems related to divergence in a variety of theoretical formulas. The direction originated with the elaboration of distribution theory was further developed by creation of the theory of hypernumbers and extrafunctions. Theory of linear hyperspaces is the next stage in this direction.