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  "body": "\n## Definition\n\nLet $M$ be a closed smooth manifold and let $E$, $F$ be smooth complex [[vector bundles|P4DZM975BV]] over $M$.\nLet $P \\colon C^\\infty(M; E) \\to C^\\infty(M; F)$ be an [[elliptic differential operator|MT9P2X85V8]].\nThen $\\ker P$ and $\\operatorname{coker} P = C^\\infty(M; F) / P(C^\\infty(M; E))$ are finite-dimensional.\nThe *analytic index* of $P$ is the integer\n$$\\operatorname{ind} P = \\dim \\ker P - \\dim \\operatorname{coker} P.$$\n\n## Remarks\n\nIf $E$ and $F$ carry Hermitian metrics and $M$ carries a smooth positive density, then $\\operatorname{coker} P$ is isomorphic to $\\ker P^*$, where $P^*$ is the formal adjoint, so $\\operatorname{ind} P = \\dim \\ker P - \\dim \\ker P^*$.\nLet $m$ be the order of $P$ and let $H^s$ denote Sobolev spaces.\nFor every real $s$, the extension $P \\colon H^{s}(M; E) \\to H^{s - m}(M; F)$ is a [[Fredholm operator|8HWGKWD6ZM]].\nIts [[Fredholm index|2V9Q4KHY1T]] equals $\\operatorname{ind} P$.\nThe index depends only on the homotopy class of the principal symbol of $P$ through elliptic symbols.\n\nExample: on $S^1 = \\R / 2\\pi\\Z$, let $P = d/d\\theta$ on complex functions.\nThe kernel is the constant functions, of dimension $1$.\nThe image is the functions with $\\int_0^{2\\pi} f \\, d\\theta = 0$, so the cokernel has dimension $1$, and $\\operatorname{ind} P = 0$.\n\nNon-example: the zero operator $C^\\infty(S^1) \\to C^\\infty(S^1)$, viewed as an operator of order $1$, is not elliptic, since its principal symbol vanishes.\nIts kernel and cokernel are infinite-dimensional, so it has no analytic index.\n",
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