Definition:Bilinear Functional

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

Let $S$ be a set of ordered doubles $\paren{y_1,y_2}$.

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

$\forall\paren{y_1,y_2}\in S:\exists x\in\R:B\sqbrk{y_1,y_2}=x$

Let $B$ be linear with respect to $y_1$ for fixed $y_2$, and linear with respect to $y_2$ 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 $B\sqbrk{y_1,y_2}$