Definition:Supremum Operator Norm

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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 $\norm {\, \cdot \,} : \map C {X, Y} \to \R$ be the mapping defined by:

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

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

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.3$: The normed space $\map {CL} {X,Y}$. Operator norm and the normed space $\map {CL} {X, Y}$