Hypernorm on hypervector spaces over a hyperfield

Hypernorm is a generalization of the notion of a norm on a vector space over a field. In this paper, we consider a hypervector space (V, +) ( V , + ) over a hyperfield, where $$+$$ + is a hyperoperation, and prove that the hypernorm is continuous. We show that the natural linear transformation from V V to V/Z V Z is continuous and open for all closed subhyperspaces Z of V V . We prove BL(V,W), B L ( V , W ) , the set of all bounded linear transformations from V V to W W is a hyper-Banach space whenever W W is complete. Furthermore, we obtain that in a hyper-Banach space V V if μ ₙ { μ n } is a sequence of continuous linear transformations with /μ ₙ(u)/ { / μ n ( u ) / } is bounded for every u ∈ V, u ∈ V , then μ ₙ { ‖ μ n ‖ } is bounded. In the sequel, we prove several properties of hypernorm and linear transformations on hypernormed spaces.