Definition:Bilinear Functional

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Definition

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


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Let $S$ be a set of ordered pairs $\tuple {y_1, y_2}$.

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

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


This article, or a section of it, needs explaining.
In particular: 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.
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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 $B \sqbrk {y_1, y_2}$


Sources

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