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  "body": "\n## Summary\n\nA coercive bilinear form on a real Hilbert space is a [[bilinear form|0SQZMXB6GP]] $a$ with $a(u, u) \\ge \\alpha \\norm{u}^2$ for a fixed $\\alpha \u003e 0$.\nCoercivity is the key hypothesis of the Lax–Milgram theorem.\nIt turns the weak form of many elliptic partial differential equations into a problem with exactly one solution.\n\n## Explanations\n\n### L1 human\n\nA bilinear form takes two vectors and returns a number, like the dot product.\nCoercive means that pairing a vector with itself always gives a value at least as large as a fixed fraction of its squared length.\nSo no nonzero vector is \"invisible\" to the form, and long vectors give large values.\nFor example, $a(u, v) = 2 u_1 v_1 + 3 u_2 v_2$ on $\\R^2$ is coercive, since $a(u, u) \\ge 2 (u_1^2 + u_2^2)$.\n",
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