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  "body": "\n## Summary\n\nA Nemytskii operator, also called a superposition operator, turns a function $g(x, u)$ of two variables into a map on functions.\nIt sends $u$ to the function $x \\mapsto g(x, u(x))$.\nIt is the simplest kind of nonlinear operator on function spaces, since it acts point by point.\nNonlinear integral equations such as the Hammerstein equation are compositions of a linear integral operator with a Nemytskii operator.\nMuch of the theory is about when such an operator maps one [[$L^p$ space|SASP14MA1Z]] into another and is continuous.\n\n## Explanations\n\n### L1 human\n\nTake a function $u$ and apply a fixed rule, such as squaring or taking the sine, to its value at every point.\nThe result is a new function, and the map from $u$ to this new function is a Nemytskii operator.\nFor example, with $g(x, u) = u^2$, the function $u(x) = x$ is sent to $x^2$.\nThe rule may also depend on the point, as in $g(x, u) = x \\sin u$.\n",
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