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.

1. We have ${}\Vert {v}\Vert \geq 0$ for all ${}v\in V$.
2. We have ${}\Vert {v}\Vert =0$ if and only if ${}v=0$.
3. For ${}\lambda \in {\mathbb {K} }$ and ${}v\in V$, we have

   ${}\Vert {\lambda v}\Vert =\vert {\lambda }\vert \cdot \Vert {v}\Vert \,.$

4. 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 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.