Definition:Bilinear Functional

Definition
Let $ y_1$, $ y_2$, $ z$ be mappings, belonging to some normed linear space.

Let $S$ be a set of ordered doubles $\left({ y_1, y_2 } \right)$.

Let $ B:S \to \R$ be a mapping:


 * $\forall \left({ y_1, y_2 } \right) \in S: \exists x \in \R: B \left[{ y_1, y_2 } \right] = x$

Let $B$ be linear $y_1$ for fixed $y_2$, and linear  $y_2$ for fixed $y_1$:


 * $ B \left [ { \alpha y_1 + \beta z, y_2} \right]= \alpha B \left [ { \alpha y_1, y_2} \right]+ \beta B \left[{ z, y_2} \right]$


 * $ B \left [ { y_1, \alpha y_2 + \beta z } \right]= \alpha B \left [ { y_1, y_2} \right]+ \beta B \left[{ y_1, z} \right]$

where $ \alpha$, $ \beta \in \R$.

Then $ B: S \to \R$ is known as a bilinear functional, denoted by $ B \left[ { y_1, y_2 } \right]$