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. In fact, a bilinear form can take values in any vector space, since the axioms make sense as long as vector addition and scalar multiplication are defined.
