\begin{definition}[Definition:Bilinear Functional]
Let $y_1$, $y_2$, $z$ be mappings, belonging to some normed linear space.
{{explain|See talk page.}}
Let $S$ be a set of ordered pairs $\tuple {y_1, y_2}$.
Let $B: S \to \R$ be a mapping defined as:
:$\forall \tuple {y_1, y_2} \in S: \exists x \in \R: B \sqbrk {y_1, y_2} = x$
{{explain|Establish the precise meaning of $B \sqbrk {y_1, y_2}$, and see whether it actually might mean the same as $\map B {y_1, y_2}$, in which case use the latter. If not, explain what it does mean.}}
Let $B$ be linear {{WRT}} $y_1$ for fixed $y_2$, and linear {{WRT}} $y_2$ for fixed $y_1$:
:$B \sqbrk {\alpha y_1 + \beta z, y_2} = \alpha B \sqbrk {\alpha y_1, y_2} + \beta B \sqbrk {z, y_2}$
:$B \sqbrk {y_1, \alpha y_2 + \beta z} = \alpha B \sqbrk {y_1, y_2} + \beta B \sqbrk {y_1, z}$
where $\alpha, \beta \in \R$.
Then $B: S \to \R$ is known as a '''bilinear functional''', denoted by $B \sqbrk {y_1, y_2}$
\end{definition}