# Definition:Norm/Bounded Linear Transformation

## Definition

Let $\HH$ and $\KK$ be Hilbert spaces.

Let $A: \HH \to \KK$ be a bounded linear transformation.

### Definition 1

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

- $\norm A = \sup \set {\norm {A h}_\KK: \norm h_\HH \le 1}$

### Definition 2

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

- $\norm A = \sup \set {\dfrac {\norm {A h}_\KK} {\norm h_\HH}: h \in \HH, h \ne \mathbf 0_\HH}$

### Definition 3

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

- $\norm A = \sup \set {\norm {A h}_\KK: \norm h_\HH = 1}$

### Definition 4

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

- $\norm A = \inf \set {c > 0: \forall h \in \HH: \norm {A h}_\KK \le c \norm h_\HH}$

## 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, $\HH = \KK$).

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
- 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*$\S \text {II}.1$