vector p-normVectorPnorm2001-10-06 03:09:312006-10-13 15:19:14DefinitionManhattan metricTaxicabL^1 normL^1 metricL^2 metricL^2 normL^\infty norm\usepackage{graphicx}
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\newcommand{\figuraex}[2]{\begin{center}\includegraphics[#2]{#1}\end{center}}A class of vector norms, called a $p$-norm and denoted $||\cdot||_p$, is defined as
\begin{displaymath}
||\,x\,||_p = (|x_1|^p + \cdots + |x_n|^p)^\frac{1}{p}\qquad p\geq1, x\in\R^n
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The most widely used are the 1-norm, 2-norm, and $\infty$-norm:
\begin{eqnarray*}
||\,x\,||_1 & =& |x_1| + \cdots + |x_n| \\
||\,x\,||_2 & =& \sqrt{|x_1|^2 + \cdots + |x_n|^2} = \sqrt{x^Tx} \\
||\,x\,||_\infty & =& \displaystyle\max_{1\leq i\leq n}|x_i|
\end{eqnarray*}
The 2-norm is sometimes called the Euclidean vector norm, because
$||\,x-y\,||_2$ yields the Euclidean distance between any two vectors $x,y\in \R^n$. 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 $\R^n$) the three mentioned norms are \PMlinkid{equivalent}{4312}. 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 \PMlinkname{$L^p$-norm}{LpSpace} in function spaces is a generalization of these norms by using counting measure.