Definition:Norm/Bounded Linear Transformation/Definition 2

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

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

Let $A: X \to Y$ be a bounded linear transformation.

The norm of $A$ is the real number defined and denoted as:


 * $\norm A = \sup \set {\dfrac {\norm {A x}_Y} {\norm x_X}: x \in X, x \ne \mathbf 0_X}$

This supremum is to be taken in $\closedint 0 \infty$ so that $\sup \O = 0$.

Also see

 * Definition:Bounded Linear Transformation