Such a function is denoted as . Particular norms are distinguished by subscripts, such
as , when referring to a norm in the space . A unit vector with respect to the norm is a vector satisfying
A vector norm on a complex vector space is defined similarly.
A common (and useful) example of a real norm is the Euclidean norm given by defined on . Note, however, that there exists vector spaces which are metric, but upon which it is not possible to define a norm. If it possible, the space is called a normed vector space. Given a metric on the vector space, a necessary and sufficient condition for this space to be a normed space, is
But given a norm, a metric can always be defined by the equation . Hence every normed space is a metric space.
|Date of creation||2013-03-22 11:43:00|
|Last modified on||2013-03-22 11:43:00|
|Last modified by||mike (2826)|
|Defines||normed vector space|