# Definition:Norm

## Abstract Algebra

### 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**:

\((N1)\) | $:$ | Positive Definiteness: | \(\displaystyle \forall x \in R:\) | \(\displaystyle \norm x = 0 \) | \(\displaystyle \iff \) | \(\displaystyle x = 0_R \) | ||

\((N2)\) | $:$ | Submultiplicativity: | \(\displaystyle \forall x, y \in R:\) | \(\displaystyle \norm {x \circ y} \) | \(\displaystyle \le \) | \(\displaystyle \norm x \times \norm y \) | ||

\((N3)\) | $:$ | 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**:

\((N1)\) | $:$ | Positive Definiteness: | \(\displaystyle \forall x \in R:\) | \(\displaystyle \norm x = 0 \) | \(\displaystyle \iff \) | \(\displaystyle x = 0_R \) | ||

\((N2)\) | $:$ | Multiplicativity: | \(\displaystyle \forall x, y \in R:\) | \(\displaystyle \norm {x \circ y} \) | \(\displaystyle = \) | \(\displaystyle \norm x \times \norm y \) | ||

\((N3)\) | $:$ | 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:

\((N1)\) | $:$ | Positive definiteness: | \(\displaystyle \forall x \in V:\) | \(\displaystyle \norm x = 0 \) | \(\displaystyle \iff \) | \(\displaystyle x = \mathbf 0_V \) | ||

\((N2)\) | $:$ | Positive homogeneity: | \(\displaystyle \forall x \in V, \lambda \in R:\) | \(\displaystyle \norm {\lambda x} \) | \(\displaystyle = \) | \(\displaystyle \norm {\lambda}_R \times \norm x \) | ||

\((N3)\) | $:$ | 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:

- $(1): \quad \norm{A} = \sup \set{\norm{Ah}_K: \norm{h}_H \le 1}$
- $(2): \quad \norm{A} = \sup \set{\dfrac {\norm{Ah}_K} {\norm{h}_H}: h \in H, h \ne \mathbf{0}_H}$
- $(3): \quad \norm{A} = \sup \set{\norm{Ah}_K: \norm{h}_H \le 1}$
- $(4): \quad \norm{A} = \inf \set{c > 0: \forall h \in H: \norm{Ah}_K \le c \norm{h}_H}$

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.

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:

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