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  "body": "\n## Definition\n\nLet $H$ be a real [[Hilbert space|HM2HHE3J51]] with norm $\\norm{\\cdot}$.\nA [[bilinear form|0SQZMXB6GP]] $a \\colon H \\times H \\to \\R$ is *coercive* if there is a constant $\\alpha \u003e 0$ such that\n$$a(u, u) \\ge \\alpha \\norm{u}^2 \\quad \\text{for all } u \\in H.$$\n\n## Remarks\n\nExample: the inner product $a(u, v) = \\ip{u}{v}$ is coercive with $\\alpha = 1$, since $a(u, u) = \\norm{u}^2$.\nNon-example: on $\\R^2$, the form $a(u, v) = u_1 v_1$ is not coercive, because $a(u, u) = 0$ for $u = (0, 1)$, while $\\norm{u}^2 = 1$.\nThe same definition is used on a real normed space.\nOn a complex Hilbert space, a sesquilinear form $a$ is called coercive if $\\operatorname{Re} a(u, u) \\ge \\alpha \\norm{u}^2$ for all $u$.\nSome authors call a coercive form elliptic.\nIn the Lax–Milgram theorem the form is also assumed bounded: $\\abs{a(u, v)} \\le M \\norm{u} \\norm{v}$ for some constant $M$ and all $u, v \\in H$.\n",
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