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  "body": "\n## Definition\n\nLet $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|0VKD1TP7F3]].\nA map $f \\colon X \\to Y$ is *nonexpansive* if\n$$d_Y(f(x), f(x')) \\le d_X(x, x') \\quad \\text{for all } x, x' \\in X.$$\n\n## Remarks\n\nEquivalently, $f$ is [[Lipschitz|0AW1E6YTJY]] with Lipschitz constant at most $1$.\nIn operator theory, $X = Y = C$ is often a subset of a Banach space with the norm distance, and the condition reads $\\norm{f(x) - f(x')} \\le \\norm{x - x'}$.\n\nExample: let $H$ be a real Hilbert space, $C \\subseteq H$ a nonempty closed convex set, and $P_C(x)$ the nearest point of $C$ to $x$.\nThe variational inequality $\\ip{x - P_C x}{c - P_C x} \\le 0$ for $c \\in C$ gives $\\norm{P_C x - P_C y}^2 \\le \\ip{x - y}{P_C x - P_C y} \\le \\norm{x - y}\\norm{P_C x - P_C y}$, so $P_C$ is nonexpansive.\n\nNon-example: $f(x) = 2x$ on $\\R$ gives $\\abs{f(1) - f(0)} = 2 \u003e 1 = \\abs{1 - 0}$.\n\nEvery [[contraction mapping|0QN69YA357]] is nonexpansive, but not conversely.\nThe Browder–Göhde–Kirk theorem gives fixed points: a nonexpansive map of a nonempty bounded closed convex subset of a uniformly convex Banach space into itself has a fixed point.\n",
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