topological vector space, the balanced hull of any open neighborhood of the origin is again open. If is a Hausdorff topological vector space and if is a compact subset of then the balanced hull of is compact. If a set is closed (respectively, convex, absorbing, a neighborhood of the origin) then the same is true of its balanced core. For any subset and any scalar For any scalar This equality holds for if and only if Thus if or then for every scalar Related notions A function on a real or complex vector space is said to be a if it satisfies any of the following equivalent conditions: whenever is a scalar satisfying and whenever and are scalars satisfying and is a balanced set for every non-negative real If is a balanced function then for every scalar and vector so in particular, for every unit length scalar (satisfying ) and every Using shows that every balanced function is a symmetric function. A real-valued function is a seminorm if and only if it is a balanced sublinear function. See also References Proofs Sources Linear algebra