Definition:Sublinear Functional

Definition

Let $E$ be a vector space over $\R$.

A real-valued function $p: E \to \R$ is called a sublinear functional if and only if it satisfies:

\((1)\)	$:$		Subadditivity:	\(\ds \forall x, y \in E:\)		\(\ds \map p {x + y} \)	\(\ds \le \)	\(\ds \map p x + \map p y \)
\((2)\)	$:$		Positive Homogeneity:	\(\ds \forall x \in E, \forall \lambda \in \R_{>0}:\)		\(\ds \map p {\lambda x} \)	\(\ds = \)	\(\ds \lambda \map p x \)

Also known as

A sublinear functional is also sometimes referred to as a Minkowski functional.

However, the notion of a Minkowski functional has been defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as a more specialised concept, so will not be used here.

Also see

• Results about sublinear functionals can be found here.

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $19.1$: The Hahn-Banach Theorem: Real Case