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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $V$ be a [[vector space|000AYPA9AJ]] over $\\mathbb{K}$.\nA *functional* on $V$ is a [[function|0EPAT51DXQ]] $J \\colon V \\to \\mathbb{K}$.\nMore generally, a functional on a subset $A \\subseteq V$ is a function $J \\colon A \\to \\mathbb{K}$.\nIn the calculus of variations, $V$ is usually a space of real functions, such as $C^1([a, b])$, and $J$ is real-valued.\n\n## Remarks\n\nExample: on $V = C([0, 1])$, the map $J(y) = \\int_0^1 y(x)^2 \\, dx$ is a real functional.\nIt is not linear, since $J(2y) = 4 J(y)$ while a linear map would give $2 J(y)$.\nA functional that is linear is a [[linear form|07YQ1C78G0]]; when $V$ is normed, the continuous linear functionals form the [[dual space|07JBSZE0V4]] $V^*$.\nNon-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.\nSome authors restrict the word to linear functionals; in the calculus of variations it means any scalar-valued map as above.\n",
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