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.