# 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:

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

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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

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**bilinear functional** - 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 5.24$: Quadratic Functionals. The Second Variation of a Functional