# vector norm

A vector norm^{} on the real vector space $V$ is a function $f:V\to \mathbb{R}$ that satisfies the following properties:

$f(x)=0\iff x=0$ | ||||

$f(x)\ge 0$ | $x\in V$ | |||

$f(x+y)\le f(x)+f(y)$ | $x,y\in V$ | |||

$f(\alpha x)=|\alpha |f(x)$ | $\alpha \in \mathbb{R},x\in V$ |

Such a function is denoted as $||x||$. Particular norms are distinguished by subscripts, such
as ${||x||}_{V}$, when referring to a norm in the space $V$. A *unit vector ^{}* with respect to the norm $||\cdot ||$ is a vector $x$ satisfying
$||x||=1$.

A vector norm on a complex vector space is defined similarly.

A common (and useful) example of a real norm is the Euclidean norm given by $||x||={({x}_{1}^{2}+{x}_{2}^{2}+\mathrm{\cdots}+{x}_{n}^{2})}^{1/2}$ defined on $V={\mathbb{R}}^{n}$.
Note, however, that there exists vector spaces^{} which are metric, but upon which it is not possible to define a norm. If it possible, the space is called a normed vector space. Given a metric on the vector space, a necessary and sufficient condition for this space to be a normed space, is

$d(x+a,y+a)=$ | $d(x,y)$ | $\forall x,y,a\in V$ | ||

$d(\alpha x,\alpha y)=$ | $|\alpha |d(x,y)$ | $\forall x,y\in V,\alpha \in \mathbb{R}$ |

But given a norm, a metric can always be defined by the equation $d(x,y)=||x-y||$. Hence every normed space is a metric space.

Title | vector norm |

Canonical name | VectorNorm |

Date of creation | 2013-03-22 11:43:00 |

Last modified on | 2013-03-22 11:43:00 |

Owner | mike (2826) |

Last modified by | mike (2826) |

Numerical id | 30 |

Author | mike (2826) |

Entry type | Definition |

Classification | msc 46B20 |

Classification | msc 18-01 |

Classification | msc 20H15 |

Classification | msc 20B30 |

Related topic | Vector |

Related topic | Metric |

Related topic | Norm |

Related topic | VectorPnorm |

Related topic | NormedVectorSpace |

Related topic | MatrixNorm |

Related topic | MatrixPnorm |

Related topic | FrobeniusMatrixNorm |

Related topic | CauchySchwarzInequality |

Related topic | MetricSpace |

Related topic | VectorSpace |

Related topic | LpSpace |

Related topic | OperatorNorm |

Related topic | BoundedOperator |

Related topic | SemiNorm^{} |

Related topic | BanachSpace |

Related topic | HilbertSpace |

Related topic | UnitVector |

Defines | normed vector space |

Defines | Euclidean norm |