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  "body": "\n## Statement\n\nLet $M$ be a compact smooth manifold without boundary, with a smooth positive density.\nLet $E$ and $F$ be complex vector bundles over $M$ with Hermitian metrics.\nLet $P \\colon \\Gamma(E) \\to \\Gamma(F)$ be an [[elliptic operator|MT9P2X85V8]] of order $m$.\nLet $P^* \\colon \\Gamma(F) \\to \\Gamma(E)$ be its formal adjoint.\nIt is the [[differential operator|CGBX1N9V8H]] with $\\int_M \\ip{Pu}{v} = \\int_M \\ip{u}{P^* v}$ for all smooth sections $u$ and $v$.\nFor $s \\in \\R$, let $H^s(E)$ and $H^s(F)$ be the [[Sobolev spaces|M4CEZ5K9HP]] of sections.\n\n1. For every $s \\in \\R$, $P$ extends to a bounded operator $P_s \\colon H^{s+m}(E) \\to H^s(F)$, and $P_s$ is a [[Fredholm operator|8HWGKWD6ZM]].\n2. The kernel of $P_s$ consists of smooth sections and equals the kernel of $P$ on $\\Gamma(E)$.\nIn particular it is finite-dimensional and does not depend on $s$.\n3. The kernel of $P^*$ is a finite-dimensional space of smooth sections.\nThe range of $P_s$ is the set of $f \\in H^s(F)$ with $\\int_M \\ip{f}{v} = 0$ for all $v$ in the kernel of $P^*$.\n4. The [[index|2V9Q4KHY1T]] of $P_s$ equals $\\dim \\ker P - \\dim \\ker P^*$ and does not depend on $s$.\n\n## Remarks\n\nThe proof constructs a [[parametrix|Z97HVFD6JE]] $Q$, a pseudodifferential operator of order $-m$ built from the inverse of the principal symbol, with $QP - I$ and $PQ - I$ smoothing.\nSmoothing operators map $H^s$ into $H^{s+1}$, and by the Rellich lemma the inclusion $H^{s+1} \\to H^s$ is [[compact|M2NEP1K5AE]] on a compact manifold.\nSo $P_s$ is invertible modulo compact operators, which is Atkinson's characterisation of Fredholm operators.\nThe parametrix also gives [[elliptic regularity|2PVQY9PCT4]]: if $Pu$ is smooth then $u$ is smooth, which yields part 2.\n\nExample: on the circle $S^1 = \\R / 2\\pi\\Z$, let $P = -d^2/dx^2$.\nIts kernel and the kernel of $P^* = P$ are the constant functions, so the index is $1 - 1 = 0$, and the range consists of the functions with average zero.\n\nCompactness of $M$ is needed.\nOn $\\R$, the operator $d/dx \\colon H^1(\\R) \\to L^2(\\R)$ is elliptic and injective, but its range is dense and not closed, so it is not Fredholm.\n\n## Explanations\n\n### L1 human\n\nAn elliptic operator on a closed manifold behaves almost like an invertible matrix.\nIt kills only a finite-dimensional space of sections, and it misses only a finite-dimensional space of targets.\nThe difference of these two dimensions, the index, is a stable number that does not change when the operator is deformed through elliptic operators.\n\n### L2 human\n\nThis theorem is the analytic foundation of index theory.\nThe index depends only on the homotopy class of the principal symbol through invertible symbols, which is why it can be computed topologically by the Atiyah–Singer index theorem.\nIt also underlies Hodge theory: applied to the Hodge Laplacian, it gives finite-dimensional spaces of harmonic forms and the orthogonal decomposition of forms.\n",
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