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 $\GF \in \set {\R, \C}$.

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.

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

Bounded Linear Functionals
Let $\GF$ be a subfield of $\C$.

Let $\struct {V, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.

Let $L : V \to \GF$ be a bounded linear functional.

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.

$p$-adic Norm on the $p$-adic Numbers
The $p$-adic norm on the $p$-adic numbers $\Q_p$ is an example of a norm on a division ring.

Also see

 * Definition:Field Norm a similar but subtly different concept