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  "body": "\n## Summary\n\nThe analytic index of an elliptic operator $P$ on a closed manifold is $\\dim \\ker P - \\dim \\operatorname{coker} P$.\nBoth dimensions are finite because elliptic operators on closed manifolds are [[Fredholm operators|8HWGKWD6ZM]] between suitable Sobolev spaces.\nSo the analytic index is a [[Fredholm index|2V9Q4KHY1T]].\nThe Atiyah–Singer index theorem computes it as the [[topological index|QDCD33CS21]].\n\n## Explanations\n\n### L1 human\n\nThe kernel of $P$ counts the independent solutions of $Pu = 0$.\nThe cokernel counts the independent conditions that $f$ must satisfy for $Pu = f$ to be solvable.\nThe analytic index is the difference of these two counts.\nTake the Cauchy–Riemann operator $\\bar\\partial$ on functions on a closed Riemann surface of genus $g$.\nIts kernel is the constants and its cokernel has dimension $g$, so the index is $1 - g$.\n",
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