A class of vector norms, called a -norm and denoted , is defined as
The most widely used are the 1-norm, 2-norm, and -norm:
The 2-norm is sometimes called the Euclidean vector norm, because 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 http://planetmath.org/node/4312equivalent. 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.
|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)|
|Author||Andrea Ambrosio (7332)|
|Synonym||Euclidean vector norm|
|Synonym||vector Euclidean norm|