# 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 $\paren{y_1,y_2}$.

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 $B\sqbrk{y_1,y_2}$

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 5.24$: Quadratic Functionals. The Second Variation of a Functional