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

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 . 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 ) the three mentioned norms are equivalent. 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 in function spaces is a generalization of these norms by using counting measure.