Definition:Norm/Bounded Linear Transformation/Definition 1

From ProofWiki
Jump to navigation Jump to search

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