# Definition:Norm/Bounded Linear Transformation

*This page is about Norm on Bounded Linear Transformation. For other uses, see Norm.*

## Contents

## Definition

Let $H, K$ be Hilbert spaces, and let $A: H \to K$ be a bounded linear transformation.

Then the **norm** of $A$, denoted $\norm A$, is the real number defined by:

- $(1): \quad \norm A = \sup \set {\norm {A h}_K: \norm h_H \le 1}$
- $(2): \quad \norm A = \sup \set {\dfrac {\norm {A h}_K} {\norm h_H}: h \in H, h \ne \mathbf 0_H}$
- $(3): \quad \norm A = \sup \set {\norm {A h}_K: \norm h_H \le 1}$
- $(4): \quad \norm A = \inf \set {c > 0: \forall h \in H: \norm {A h}_K \le c \norm h_H}$

These definitions are equivalent, as proved in Equivalence of Definitions of Norm of Linear Transformation.

An important property of $\norm A$ is that:

- $\forall h \in H: \norm {A h}_K \le \norm A \norm h_H$

This is proven on Submultiplicativity of Operator Norm.

As $A$ is bounded, it is assured that $\norm A < \infty$.

### Operator Norm

Above definition also applies when in fact $A$ is a linear operator (that is, $H = K$).

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

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

## Also see

- Definition:Hilbert Space
- Definition:Bounded Linear Transformation
- Equivalence of Definitions of Norm of Linear Transformation
- Definition:Norm on Bounded Linear Functional, a special case where $K$ is in fact the ground field of $H$.

## Sources

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