Definition:Minkowski Functional/Normed Vector Space

Definition
Let $\Bbb F \in \set {\R, \C}$.

Let $\struct {X, \norm \cdot}$ be a normed vector space over $\Bbb F$.

Let $C$ be an open convex subset of $X$ with $0 \in C$.

We define the Minkowski functional of $C$, $p_C : X \to \hointr 0 \infty$ by:


 * $\map {p_C} x = \inf \set {t > 0 : t^{-1} x \in C}$

for each $x \in X$.

Also see

 * Minkowski Functional of Open Convex Set in Normed Vector Space is Well-Defined