# vector p-norm

A class of vector norms, called a $p$-norm and denoted $||\cdot |{|}_{p}$, is defined as

$${||x||}_{p}={({|{x}_{1}|}^{p}+\mathrm{\cdots}+{|{x}_{n}|}^{p})}^{\frac{1}{p}}\mathit{\hspace{1em}\hspace{1em}}p\ge 1,x\in {\mathbb{R}}^{n}$$ |

The most widely used are the 1-norm, 2-norm, and $\mathrm{\infty}$-norm:

${||x||}_{1}$ | $=$ | $|{x}_{1}|+\mathrm{\cdots}+|{x}_{n}|$ | ||

${||x||}_{2}$ | $=$ | $\sqrt{{|{x}_{1}|}^{2}+\mathrm{\cdots}+{|{x}_{n}|}^{2}}=\sqrt{{x}^{T}x}$ | ||

${||x||}_{\mathrm{\infty}}$ | $=$ | $\underset{1\le i\le n}{\mathrm{max}}|{x}_{i}|$ |

The 2-norm is sometimes called the Euclidean vector norm, because ${||x-y||}_{2}$ yields the Euclidean distance between any two vectors $x,y\in {\mathbb{R}}^{n}$. The 1-norm is also called the taxicab metric (sometimes Manhattan metric) since the distance of two points can be viewed as the distance a taxi would travel on a city (horizontal and vertical movements).

A useful fact is that for finite dimensional spaces (like ${\mathbb{R}}^{n}$) the three mentioned norms are http://planetmath.org/node/4312equivalent^{}. Moreover, all $p$-norms are equivalent. This can be proved using that any norm has to be continuous^{} in the $2$-norm and working in the unit circle.

The ${L}^{p}$-norm (http://planetmath.org/LpSpace) in function spaces is a generalization^{} of these norms by using counting measure.

Title | vector p-norm |

Canonical name | VectorPnorm |

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

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

Owner | Andrea Ambrosio (7332) |

Last modified by | Andrea Ambrosio (7332) |

Numerical id | 21 |

Author | Andrea Ambrosio (7332) |

Entry type | Definition |

Classification | msc 46B20 |

Classification | msc 05Cxx |

Classification | msc 05-01 |

Classification | msc 20H15 |

Classification | msc 20B30 |

Synonym | Minkowski norm |

Synonym | Euclidean vector norm |

Synonym | vector Euclidean norm |

Synonym | vector 1-norm |

Synonym | vector 2-norm |

Synonym | vector infinity-norm |

Synonym | L^p metric |

Synonym | L^p |

Related topic | VectorNorm |

Related topic | CauchySchwartzInequality |

Related topic | HolderInequality |

Related topic | FrobeniusMatrixNorm |

Related topic | LpSpace |

Related topic | CauchySchwarzInequality |

Defines | Manhattan metric |

Defines | Taxicab |

Defines | L^1 norm |

Defines | L^1 metric |

Defines | L^2 metric |

Defines | L^2 norm |

Defines | L^∞norm |