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  "body": "\n## Summary\n\nA functional on a normed space is coercive if its values tend to $+\\infty$ as the norm of the argument tends to infinity.\nThen every minimising sequence is bounded, which is the compactness step of the [[direct method|TE379HBTY0]].\nIn reflexive spaces bounded sequences have weakly convergent subsequences, so coercivity and weak lower semicontinuity together give a minimiser.\nThe word is related to, but different from, a [[coercive bilinear form|86KRHS6X22]].\n\n## Explanations\n\n### L1 human\n\nA coercive function grows without bound in every direction far from the origin.\nSo its low values all lie in a bounded region, and a search for the minimum cannot escape to infinity.\nOn the real line, $f(x) = x^2$ is coercive, while $f(x) = e^x$ is not, since it stays small as $x \\to -\\infty$.\nThe function $e^x$ indeed has no minimum: its infimum $0$ is approached only far away.\n",
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