Definition:Bilinear Form

Definition
Let $R$ be a ring.

Let $R_R$ denote the $R$-module $R$.

Let $M_R$ be an $R$-module.

A bilinear form on $M_R$ is a bilinear mapping $B : M_R \times M_R \to R_R$.

In the context of calculus of variations
Let $B$ be a bilinear functional.

Let $B$ be defined on a finite-dimensional space.

Then $B$ is a bilinear form.

Also known as
It is usual to gloss over the modular nature of $R_R$ and consider $B$ merely as a mapping from the $R$-module $M$ directly to the ring $R$:

Hence in this manner, a bilinear form on $M$ is a defined as a bilinear mapping $B : M \times M \to R$.

Also see

 * Definition:Relative Matrix of Bilinear Form
 * Definition:Quadratic Form
 * Definition:Associated Quadratic Form
 * Definition:Bilinear Space
 * Definition:Linear Form