Definition:Norm

Definition
A norm is a measure which describes a sense of the size or length of a mathematical object.

In its various contexts:

Unital Algebra
Let $R$ be a division ring with norm $\norm {\,\cdot\,}_R$.

Bounded Linear Transformations
Let $\HH$ and $\KK$ be Hilbert spaces.

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

The norm on the vector space of bounded linear transformations is an example of a norm on a vector space.

Bounded Linear Functionals
Let $H$ be a Hilbert space, and let $L$ be a bounded linear functional on $H$.

The norm on the vector space of bounded linear functionals is an example of a norm on a vector space.

Real Numbers
The absolute value function on the real numbers $\R$ is an example of a norm on a division ring.

Complex Numbers
The (complex) modulus function on the complex numbers $\C$ is an example of a norm on a division ring.

Quaternions
The (quaternion) modulus function on the quaternions $\mathbb H$ is an example of a norm on a non-commutative division ring.

$p$-adic Norm on the Rationals
The $p$-adic norm on the rational numbers $\Q$ is an example of a norm on a division ring.

Also see

 * Definition:Field Norm a similar but subtly different concept