A bilinear form on a real vector space is a function
 |
that satisfies the following axioms for any scalar 
and any choice of vectors 
, 
, 
,
, 
, and 
.
1. 
2. 
3. 
.
For example, the function 
is a bilinear form on 
.
On a complex vector space, a bilinear form takes values in the complex numbers. More generally,
a bilinear form on a vector space over a field 
takes values in 
. Allowing values in another vector
space over the same field 
gives a bilinear function.
The same axioms apply because vector
addition and scalar multiplication are
defined in the target vector space.
