Definition:Norm
Definition
A norm is a generalization of the concept of complex modulus and absolute value.
Hence it 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: | \(\ds \forall x, y \in V:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |
Matrix Space
Let $m, n \in \Z_{>0}$ be (strictly) positive integers.
Let $\map {\MM_\GF} {m, n}$ denote the vector space of matrices of order $m \times n$ over a field $\GF$.
A norm over $\map {\MM_\GF} {m, n}$ is known as a matrix norm.
A matrix norm on $\map {\MM_\GF} {m, n}$ is a map from $\map {\MM_\GF} {m, n}$ to the nonnegative reals:
- $\norm {\, \cdot \,}: \map {\MM_\GF} {m, n} \to \R_{\ge 0}$
satisfying the (matrix) norm axioms:
\((\text N 1)\) | $:$ | Positive Definiteness: | \(\ds \forall \mathbf A \in \map {\MM_\GF} {m, n}:\) | \(\ds \norm {\mathbf A} = 0 \) | \(\ds \iff \) | \(\ds \mathbf A = \mathbf 0_{m, n} \) | where $\mathbf 0_{m, n}$ denotes the zero matrix of order $m \times n$ | ||
\((\text N 2)\) | $:$ | Positive Homogeneity: | \(\ds \forall x \in \map {\MM_\GF} {m, n}, \lambda \in \GF:\) | \(\ds \norm {\lambda \mathbf A} \) | \(\ds = \) | \(\ds \norm \lambda \times \norm {\mathbf A} \) | where $\norm \lambda$ denotes the (division ring) norm of $\lambda$ | ||
\((\text N 3)\) | $:$ | Triangle Inequality: | \(\ds \forall \mathbf A, \mathbf B \in \map {\MM_\GF} {m, n}:\) | \(\ds \norm {\mathbf A + \mathbf B} \) | \(\ds \le \) | \(\ds \norm {\mathbf A} + \norm {\mathbf B} \) |
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 a 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
- Results about norm theory can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): norm
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): norm
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): norm