normed vector space
Let be a field which is either or . A over is a pair where is a vector space over and is a function such that
for all and if and only if in (positive definiteness)
for all and all
for all (the triangle inequality)
The function is called a norm on .
Some properties of norms:
In this metric, the norm defines a continuous map from to - this is an easy consequence of the triangle inequality.
If is an inner product space, then there is a natural induced norm given by for all .
|Title||normed vector space|
|Date of creation||2013-03-22 12:13:45|
|Last modified on||2013-03-22 12:13:45|
|Last modified by||rspuzio (6075)|
|Synonym||normed linear space|
|Defines||metric induced by a norm|
|Defines||metric induced by the norm|