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:

Ring

Let $\struct {R, +, \circ}$ be a ring whose zero is denoted $0_R$.

A (submultiplicative) norm on $R$ is a mapping from $R$ to the non-negative reals:

$\norm {\,\cdot\,}: R \to \R_{\ge 0}$

satisfying the (ring) submultiplicative norm axioms:

\((\text N 1)\)	$:$		Positive Definiteness:	\(\ds \forall x \in R:\)		\(\ds \norm x = 0 \)	\(\ds \iff \)	\(\ds x = 0_R \)
\((\text N 2)\)	$:$		Submultiplicativity:	\(\ds \forall x, y \in R:\)		\(\ds \norm {x \circ y} \)	\(\ds \le \)	\(\ds \norm x \times \norm y \)
\((\text N 3)\)	$:$		Triangle Inequality:	\(\ds \forall x, y \in R:\)		\(\ds \norm {x + y} \)	\(\ds \le \)	\(\ds \norm x + \norm y \)

Division Ring

Let $\struct {R, +, \circ}$ be a division ring whose zero is denoted $0_R$.

A (multiplicative) norm on $R$ is a mapping from $R$ to the non-negative reals:

$\norm {\,\cdot\,}: R \to \R_{\ge 0}$

satisfying the (ring) multiplicative norm axioms:

\((\text N 1)\)	$:$		Positive Definiteness:	\(\ds \forall x \in R:\)		\(\ds \norm x = 0 \)	\(\ds \iff \)	\(\ds x = 0_R \)
\((\text N 2)\)	$:$		Multiplicativity:	\(\ds \forall x, y \in R:\)		\(\ds \norm {x \circ y} \)	\(\ds = \)	\(\ds \norm x \times \norm y \)
\((\text N 3)\)	$:$		Triangle Inequality:	\(\ds \forall x, y \in R:\)		\(\ds \norm {x + y} \)	\(\ds \le \)	\(\ds \norm x + \norm y \)

Vector Space

Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$.

Let $V$ be a vector space over $R$, with zero $0_V$.

A norm on $V$ is a map from $V$ to the nonnegative reals:

$\norm{\,\cdot\,}: V \to \R_{\ge 0}$

satisfying the (vector space) norm axioms:

\((\text N 1)\)	$:$		Positive Definiteness:	\(\ds \forall x \in V:\)		\(\ds \norm x = 0 \)	\(\ds \iff \)	\(\ds x = \mathbf 0_V \)
\((\text N 2)\)	$:$		Positive Homogeneity:	\(\ds \forall x \in V, \lambda \in R:\)		\(\ds \norm {\lambda x} \)	\(\ds = \)	\(\ds \norm {\lambda}_R \times \norm x \)
\((\text N 3)\)	$:$		Triangle Inequality:	\(\ds \forall x, y \in V:\)		\(\ds \norm {x + y} \)	\(\ds \le \)	\(\ds \norm x + \norm y \)

Algebra

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

Let $A$ be an algebra over $R$.

A norm on $A$ is a vector space norm $\norm{\,\cdot\,}: A \to \R_{\ge 0}$ on $A$ as a vector space such that:

for all $a, b \in A: \norm {a b} \le \norm a \cdot \norm b$

Unital Algebra

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

Let $A$ be an unital algebra over $R$ with unit $e$.

A norm on $A$ is an algebra norm $\norm {\,\cdot\,}: A \to \R_{\ge 0}$ such that:

$\norm e = 1$

Examples in Functional Analysis

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

$\norm A = \sup \set {\norm {A x}_Y : \norm x_X \le 1}$

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 of $L$ is defined as the supremum:

$\norm L = \sup \set {\size {L v}: \norm v \le 1}$

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

Examples In Analysis

Real Numbers

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

Let $x \in \R$ be a real number.

The absolute value of $x$ is denoted $\size x$, and is defined using the usual ordering on the real numbers as follows:

$\size x = \begin{cases} x & : x > 0 \\ 0 & : x = 0 \\ -x & : x < 0 \end{cases}$

Complex Numbers

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

Let $z = a + i b$ be a complex number, where $a, b \in \R$.

The (complex) modulus of $z$ is written $\cmod z$, and is defined as the square root of the sum of the squares of the real and imaginary parts:

$\cmod z := \sqrt {a^2 + b^2}$

Quaternions

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

The (quaternion) modulus of $\mathbf x$ is the real-valued function defined and denoted as:

$\size {\mathbf x} := \sqrt {a^2 + b^2 + c^2 + d^2}$

$p$-adic Norm on the Rationals

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

Let $\nu_p: \Q \to \Z \cup \set {+\infty}$ be the $p$-adic valuation on $\Q$.

The $p$-adic norm on $\Q$ is the mapping $\norm {\,\cdot\,}_p: \Q \to \R_{\ge 0}$ defined as:

$\forall q \in \Q: \norm q_p := \begin{cases}

0 & : q = 0 \\ p^{-\map {\nu_p} q} & : q \ne 0 \end{cases}$

$p$-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.

Let $\norm {\, \cdot \,}_p:\Q_p \to \R_{\ge 0}$ be the norm on the quotient ring $\Q_p$ defined by:

$\ds \forall \eqclass{x_n}{} \in \Q_p: \norm {\eqclass{x_n}{} }_p = \lim_{n \mathop \to \infty} \norm{x_n}_p$

The norm $\norm {\,\cdot\,}_p$ on $\Q_p$ is called the $p$-adic norm on $\Q_p$.

Also see

• Definition:Field Norm a similar but subtly different concept

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): norm