PlanetMath: vector p-norm

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vector p-norm

(Definition)

A class of vector norms, called a -norm and denoted $\vert\vert\cdot\vert\vert _p$ , is defined as

$\begin{displaymath} \vert\vert\,x\,\vert\vert _p = (\vert x_1\vert^p + \cdots + \vert x_n\vert^p)^\frac{1}{p}\qquad p\geq1, x\in\mathbbmss{R}^n \end{displaymath}$

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

$\begin{eqnarray*} \vert\vert\,x\,\vert\vert _1 & =& \vert x_1\vert + \cdots + \v... ...ert _\infty & =& \displaystyle\max_{1\leq i\leq n}\vert x_i\vert \end{eqnarray*}$

The 2-norm is sometimes called the Euclidean vector norm, because $\vert\vert\,x-y\,\vert\vert _2$ yields the Euclidean distance between any two vectors $x,y\in \mathbbmss{R}^n$ . 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 $\mathbbmss{R}^n$ ) the three mentioned norms are equivalent. Moreover, all -norms are equivalent. This can be proved using that any norm has to be continuous in the -norm and working in the unit circle.

The -norm in function spaces is a generalization of these norms by using counting measure.

"vector p-norm" is owned by Andrea Ambrosio. [ full author list (3) | owner history (3) ]

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-space, Cauchy-Schwarz inequality

Other names:

Minkowski norm, Euclidean vector norm, vector Euclidean norm, vector 1-norm, vector 2-norm, vector infinity-norm, L^p metric, L^p

Also defines:

Manhattan metric, Taxicab, L^1 norm, L^1 metric, L^2 metric, L^2 norm, L^\infty norm

Attachments:: $\lim_{p \to \infty} \lVert x \rVert_p = \lVert x \rVert_{\infty}$ (Result) by Koro

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Cross-references: counting measure, function spaces, unit circle, continuous, equivalent, norms, finite dimensional, points, distance, taxicab metric, vectors, Euclidean distance, vector norms, class
There are 7 references to this entry.

This is version 9 of vector p-norm, born on 2001-10-06, modified 2006-10-13.
Object id is 92, canonical name is VectorPnorm.
Accessed 42125 times total.

Classification:

AMS MSC:

46B20 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Geometry and structure of normed linear spaces)

Pending Errata and Addenda

None.[ View all 8 ]

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