Normed vector space/K/Introduction/Section

Due to fact, the norm associated to an inner product is a norm in the sense of the following definition. In particular, a vector space with an inner product is a normed vector space.

Definition

Let ${}V$ be a ${}{\mathbb {K} }$ -vector space. A mapping

\Vert {-}\Vert \colon V\longrightarrow \mathbb {R} ,v\longmapsto \Vert {v}\Vert ,

is called norm if the following properties hold.

We have ${}\Vert {v}\Vert \geq 0$ for all ${}v\in V$ .
We have ${}\Vert {v}\Vert =0$ if and only if ${}v=0$ .
For ${}\lambda \in {\mathbb {K} }$ and ${}v\in V$ , we have
${}\Vert {\lambda v}\Vert =\vert {\lambda }\vert \cdot \Vert {v}\Vert \,.$
For ${}v,w\in V$ , we have
${}\Vert {v+w}\Vert \leq \Vert {v}\Vert +\Vert {w}\Vert \,.$

Definition

A ${}{\mathbb {K} }$ -vector space is called a normed vector space if a norm

{}\Vert {-}\Vert

is defined on it.

On a Euclidean vector space, the norm given via the inner product is also called the Euclidean norm. For ${}V=\mathbb {R} ^{n}$ , endowed with the standard inner product, we have

{}\Vert {v}\Vert ={\sqrt {\sum _{i=1}^{n}v_{i}^{2}}}\,.

Example

In ${}{\mathbb {K} }^{n}$ , taking

{}\Vert {x}\Vert _{\rm {max}}:={\max {\left(\vert {x_{i}}\vert ,i=1,\ldots ,n\right)}}\,,

a norm is defined, which is called the maximum norm.

The sum metric is also called the *taxicab metric*. The green line represents the Euclidean distance, the other paths represent the sum distance.

Example

In ${}{\mathbb {K} }^{n}$ , taking

{}\mid \mid \!x\!\mid \mid _{\operatorname {sum} }:=\sum _{i=1}^{n}\vert {x_{i}}\vert \,,

defines a norm, which is called the sum norm.

For a vector ${}v\in V$ , ${}v\neq 0$ , in a normed vector space ${}V$ , the vector ${}{\frac {v}{\Vert {v}\Vert }}$ is called the corresponding normalized. Such a normalized vector has norm ${}1$ . Passing to the normalized vector is also called normalization.