Functional

Remarks

Example: on $V = C([0, 1])$, the map $J(y) = \int_0^1 y(x)^2 \, dx$ is a real functional. It is not linear, since $J(2y) = 4 J(y)$ while a linear map would give $2 J(y)$. A functional that is linear is a linear form; when $V$ is normed, the continuous linear functionals form the dual space $V^*$. Non-example: the map $y \mapsto y'$ on $C^1([0, 1])$ takes values in $C([0, 1])$, not in $\mathbb{K}$, so it is an operator and not a functional. Some authors restrict the word to linear functionals; in the calculus of variations it means any scalar-valued map as above.