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  "body": "\n## Summary\n\nThe mutual energy of two positive measures is the double integral of a potential-theoretic kernel against one measure in each variable.\nIt is the interaction energy of two charge distributions, and it equals the integral of the potential of one measure against the other.\nThe [[energy of a measure|SFSRQCM3EG]] is the mutual energy of the measure with itself.\nSo the mutual energy is the bilinear form whose quadratic form is the energy.\nIts [[positive definiteness|PNZT35FFN2]] turns measures of finite energy into a pre-Hilbert space, which is the basis of the energy method in potential theory.\n\n## Explanations\n\n### L1 human\n\nTake two clouds of charge.\nTheir mutual energy adds up the repulsion between each bit of the first cloud and each bit of the second.\nFor two point charges at $a$ and $b$ in space, it is $1/\\abs{a - b}$, which grows as the charges approach each other.\n",
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