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

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

### 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: $\displaystyle \forall x \in V:$ $\displaystyle \norm x = 0$ $\displaystyle \iff$ $\displaystyle x = \mathbf 0_V$ $(\text N 2)$ $:$ Positive homogeneity: $\displaystyle \forall x \in V, \lambda \in R:$ $\displaystyle \norm {\lambda x}$ $\displaystyle =$ $\displaystyle \norm {\lambda}_R \times \norm x$ $(\text N 3)$ $:$ Triangle inequality: $\displaystyle \forall x, y \in V:$ $\displaystyle \norm {x + y}$ $\displaystyle \le$ $\displaystyle \norm x + \norm y$

### 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 $H, K$ 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.

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 of $L$ is the real number defined as the supremum:

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

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 $a, b \in \R$.

Then 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.

Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion, where $a, b, c, d \in \R$.

Then the (quaternion) modulus of $\mathbf x$ is written as $\vert \mathbf x \vert$ and is defined as:

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

The quaternion modulus is a real-valued function, and as when appropriate is referred to as the quaternion modulus function.

### $p$-adic Norm on the Rationals

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

## Definition

### Definition 1

Let $p \in \N$ be a prime.

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:

$\forall q \in \Q: \norm q_p := \begin{cases} 0 & : q = 0 \\ p^{-\map {\nu_p} q} & : q \ne 0 \end{cases}$

### Definition 2

Let $p \in \N$ be a prime.

Let $k, m, n \in \Z : p \nmid m, n$.

Let $\displaystyle r := p^k \frac m n$.

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

$\forall r \in \Q: \norm r_p := \begin{cases} 0 & : r = 0 \\ p^{-k} & : r \ne 0 \end{cases}$

### $p$-adic Metric

The $p$-adic metric on $\Q$ is the metric induced by $\norm{\cdot}_p$:

$\forall x, y \in \Q: \map d {x, y} = \norm{x - y}_p$

### $p$-adic Numbers

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

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