Definition:Supremum Operator Norm

Definition
Let $\struct {X, \norm {\, \cdot \,}_X}$, $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces.

Let $\map C {X, Y}$ be the continuous linear transformation space.

Let $A \in \map C {X, Y}$ be a continuous linear transformation.

Let $\norm {\, \cdot \,} : \map C {X, Y} \to \R$ be a mapping such that:


 * $\forall A \in \map C {X, Y} : \norm A := \map \sup {\norm {Ax}_Y : x \in X, \norm {x}_X \le 1}$

Then $\norm {\, \cdot \,}$ is called the supremum operator norm.