Definition:Bilinear Form

Definition
Let $R$ be a ring with unity.

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

A bilinear form on $M$ is a bilinear mapping $b : M \times M \to R$.

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.