Normed vector space/Metric/Introduction/Section

Definition

Let ${}M$ be a set. A mapping ${}d\colon M\times M\rightarrow \mathbb {R}$ is called a metric (or a distance function), if for all ${}x,y,z\in M$ the following conditions hold:

1. ${}d{\left(x,y\right)}=0$ if and only if ${}x=y$ (positivity),
2. ${}d{\left(x,y\right)}=d{\left(y,x\right)}$ (symmetry), and
3. ${}d{\left(x,y\right)}\leq d{\left(x,z\right)}+d{\left(z,y\right)}$ (triangle inequality).

A metric space is a pair ${}(M,d)$, where ${}M$ is a set and ${}d\colon M\times M\rightarrow \mathbb {R}$

is a metric.

Definition

On a normed vector space ${}V$ with norm ${}\Vert {-}\Vert$, we define the corresponding metric by

${}d(v,w):=\Vert {v-w}\Vert \,.$

This is indeed a metric.

Lemma

A normed vector space ${}V$ is with the corresponding metric a

metric space.

Proof

See exercise.

$\Box$

In particular, a Euclidean space is a metric space.

Remark

An affine space over a normed vector space ${}V$ is a metric space, be setting

${}d(P,Q)=\Vert {\overrightarrow {PQ}}\Vert \,.$

This metric is invariant under translations.

Example

Let ${}(M,d)$ be a metric space, and let ${}T\subseteq M$ denote a subset. Then ${}T$ is also a metric space by setting

${}d_{T}(x,y):=d(x,y)\,$

for all ${}x,y\in T$. This metric is called the induced metric.

Hence, every subset of an affine space over an Euclidean or normed vector space is a metric space.