Definition:Norm/Bounded Linear Transformation/Definition 1

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:


 * $\norm A = \sup \set {\norm {A x}_Y : \norm x_X \le 1}$

Also see

 * Definition:Bounded Linear Transformation