# Definition:Norm/Bounded Linear Transformation

## 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.

### Definition 1

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

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

### Definition 2

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 $\sup \O = 0$.

### Definition 3

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 $\sup \O = 0$.

### Definition 4

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

$\norm A = \inf \set {c > 0: \forall x \in X: \norm {A x}_Y \le c \norm x_X}$

## Also known as

The definition of a norm of a bounded linear transformation also applies when in fact $A$ is a linear operator (that is, $X = Y$).

Hence the norm of a bounded linear operator is also defined.

As a case of pars pro toto, the norm defined here is commonly referred to as the operator norm, even when pertaining to a linear transformation.

However, in order not to cause confusion, that usage is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$.