Definition:Norm

From ProofWiki
Jump to: navigation, search

Abstract Algebra

Ring

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


A norm on $R$ is a submultiplicative norm on $R$. That 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 norm on $R$ is a multiplicative norm on $R$. That 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 $\left\vert{x}\right\vert$, and is defined using the ordering on the real numbers as follows:

$\left\vert{x}\right\vert = \begin{cases} x & : x > 0 \\ 0 & : x = 0 \\ -x & : x < 0 \end{cases}$


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 norm of quaternion on the quaternions $\mathbb H$ is an example of a norm on a division ring.


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

Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$.


The norm of $\mathbf x$ is the real number defined as:

$n \left({\mathbf x}\right) := \left\vert{\mathbf x \overline {\mathbf x} }\right\vert = \left\vert{\overline {\mathbf x} \mathbf x }\right\vert = a^2 + b^2 + c^2 + d^2$