n-norm 范式

A class of vector norms, called a p-norm and denoted    ,  is defined as :

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

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 )

The 2-norm is sometimes called the Euclidean vector norm , because  yield the Euclidean distance between any two vectors .

A useful fact is that for finite dimensional spaces ( like R^n ) the three 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.

参考 : 

http://planetmath.org/vectorpnorm