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