Definition:Norm/Bounded Linear Transformation/Definition 3

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:

$\norm A = \sup \set {\norm {A x}_X : \norm x_X = 1}$

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

Also see

• Definition:Bounded Linear Transformation