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  "body": "\n## Statement\n\nLet $X$ be a [[Banach space|06GZ956657]], let $K \\colon X \\to X$ be a [[compact operator|M2NEP1K5AE]] such that $I - K$ is injective, and let $f \\in X$.\nLet $\\varphi \\in X$ be the solution of $\\varphi - K\\varphi = f$.\nLet $P_n \\colon X \\to X$ be bounded linear projections onto finite-dimensional subspaces $X_n$ such that $P_n \\psi \\to \\psi$ for every $\\psi \\in X$.\nThen there is $n_0$ such that for every $n \\ge n_0$ the [[projection method|2AFNVJ54G7]] equation $\\varphi_n - P_n K \\varphi_n = P_n f$ has a unique solution $\\varphi_n \\in X$, which lies in $X_n$.\nMoreover there is a constant $M$, independent of $n$ and $f$, such that\n$$\\norm{\\varphi_n - \\varphi} \\le M \\norm{P_n \\varphi - \\varphi} \\quad \\text{for all } n \\ge n_0,$$\nand in particular $\\varphi_n \\to \\varphi$.\n\n## Remarks\n\nThe solution $\\varphi$ exists because, for compact $K$, injectivity of $I - K$ implies that $I - K$ is invertible with bounded inverse (Riesz theory, the [[Fredholm alternative|4HH8C9V8JQ]]).\nSince $P_n \\psi \\to \\psi$ for each $\\psi$, the norms $\\norm{P_n}$ are bounded by the [[uniform boundedness principle|YG0042V8CJ]], so the convergence is uniform on compact sets.\nThe set $\\set{K\\psi : \\norm{\\psi} \\le 1}$ is relatively compact, hence $\\norm{P_n K - K} \\to 0$.\nA perturbation argument then shows that $I - P_n K$ is invertible for large $n$, with $\\norm{(I - P_n K)^{-1}} \\le M$.\nFinally $(I - P_n K)(\\varphi_n - \\varphi) = P_n \\varphi - \\varphi$, which gives the error bound; and if $\\varphi_n - P_nK\\varphi_n = P_n f$ then $\\varphi_n = P_n(f + K\\varphi_n) \\in X_n$.\n\nExample: for a continuous kernel on $[a, b]$, collocation with piecewise linear interpolation on grids with mesh tending to $0$ satisfies the hypothesis in $C[a, b]$, since $P_n \\psi \\to \\psi$ uniformly for continuous $\\psi$.\nThe error is then of the order of the interpolation error of the exact solution, for instance $O(h^2)$ for a twice continuously differentiable $\\varphi$ and mesh size $h$.\n\nThe hypothesis of pointwise convergence $P_n \\psi \\to \\psi$ is needed.\nFor collocation with polynomial interpolation at equidistant points in $C[a, b]$ it fails, since the norms of the interpolation operators grow without bound, and the theorem gives nothing.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that for equations of the form \"unknown minus a smoothing integral equals data\", projection methods work as soon as the projections themselves approximate every function well.\nThe error of the computed solution is then at most a fixed multiple of the best error that the projections can achieve on the exact solution.\nSo the quality of the method is reduced to a question of approximation theory.\n\n### L2 human\n\nThe key step is that a compact operator turns pointwise convergence of $P_n$ into norm convergence of $P_n K$.\nThis is why second-kind equations with compact $K$ are so well suited to numerical methods, in contrast to first-kind equations, which are ill-posed.\nIn a Hilbert space with orthogonal projections, this gives the convergence of the Galerkin method for second-kind equations.\nFor the Nyström method, $K$ is replaced by quadrature operators $K_n$ that do not converge in norm, and the same conclusion needs [[collectively compact operator approximation|AEZE6B2XXY]].\n",
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