Hypernorm on hypervector spaces over a hyperfield

Abstract

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