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  "body": "\n## Summary\n\nA functional is a map that takes an element of a vector space, usually a function, and returns a number.\nLength of a curve, energy of a field and the value of an integral are functionals of the curve, the field or the integrand.\nFunctionals are the objects that the [[calculus of variations|PQ25NHV0KS]] minimises and maximises.\nThe linear ones, the [[linear forms|07YQ1C78G0]], are the elements of the [[dual space|07JBSZE0V4]] and are central in functional analysis.\n\n## Explanations\n\n### L1 human\n\nAn ordinary function eats a number and gives a number.\nA functional eats a whole function and gives a single number.\nThink of a curve drawn between two points: its length is one number, so \"length\" is a functional on the set of curves.\nFor example, $J(y) = \\int_0^1 y(x)^2 \\, dx$ sends $y(x) = x$ to $1/3$ and $y(x) = 1$ to $1$.\n",
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