Seminorm generating relations and their Minkowski functionals (Q2714151)

In this paper the authors remark that if \(X\) is a linear space, then a multivalued mapping \(F:R_+\Rightarrow X\) (called a seminorm generating relation) has the properties: \(F(r)+F(s)\subset F(r+s)\) for all \(r,s\in R_+\); \(\lambda F(r)\subset F(tr)\) for all \(\lambda \in K\) (\(R\) or \(C\)) and \(r,t\in R_+\) with \(|\lambda|\leq t\), if and only if there exists an absorbing, balanced, convex subset \(A\) of \(X\) such that \(f(t)=tA\), for all \(t\in R_+\). Consequently, some elementary properties of these multivalued mappings can be obtained. Particularly, known properties of the Minkowski functional of an absorbing, balanced, convex set in a linear space can be formulated as properties of seminorm generating relations. However, this paper can present a didactic interest.