# 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 $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also see

- Definition:Hilbert Space
- Definition:Bounded Linear Transformation
- Definition:Norm on Bounded Linear Functional

- Norm on Bounded Linear Transformation is Finite
- Fundamental Property of Norm on Bounded Linear Transformation
- Norm on Bounded Linear Transformation is Submultiplicative

- Definition:Operator Norm: the term used for a norm when $\HH = \KK$

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*(2nd ed.) $\S \text {II}.1$