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,+) over a hyperfield, where + is a hyperoperation, and prove that the hypernorm is continuous. We show that the natural linear transformation from V to VZ is continuous and open for all closed subhyperspaces Z of V. We prove BL(V,W), the set of all bounded linear transformations from V to W is a hyper-Banach space whenever W is complete. Furthermore, we obtain that in a hyper-Banach space V if {μ_n} is a sequence of continuous linear transformations with {/μ_n(u)/} is bounded for every u∈V, then {‖μ_n‖} is bounded. In the sequel, we prove several properties of hypernorm and linear transformations on hypernormed spaces.