Locally compact multivector extensions

The aim of this paper is to develop a locally compact extension of an arbitrary normed space in such a way that the initial algebraic structure is prolonged in some sense. To obtain such an extension, we weaken vector space axioms allowing a set-valued addition and introduce in this scheme a topological structure, by means of a hypertopology, and a compatible proximity. Finally, the locally compact multivector extension appears as an ultrafilter space. We also provide a Young measure related interpretation of these extensions when the normed space is an L^p space.