Given an 
-dimensional
vector
![x=[x_1; x_2; |; x_n],](/images/equations/VectorNorm/NumberedEquation1.svg) |
(1)
|
a general vector norm 
,
sometimes written with a double bar as 
, is a nonnegative norm
defined such that
1. 
when 
and 
iff 
.
2. 
for any scalar 
.
3. 
.
In this work, a single bar is used to denote a vector norm, absolute value, or complex modulus, while a double bar is reserved for denoting a matrix norm.
The vector norm 
for 
, 2, ... is defined as
 |
(2)
|
The 
-norm of vector 
is implemented as Norm[v,
p], with the 2-norm being returned by Norm[v].
The special case 
is defined as
 |
(3)
|
The most commonly encountered vector norm (often simply called "the norm" of a vector, or sometimes the magnitude of a vector) is the L2-norm, given by
 |
(4)
|
This and other types of vector norms are summarized in the following table, together with the value of the norm for the example vector 
.
| name | symbol | value | approx. |
 -norm |  | 6 | 6.000 |
 -norm |  |  | 3.742 |
 -norm |  |  | 3.302 |
 -norm |  |  | 3.146 |
 -norm |  | 3 | 3.000 |
