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  "body": "\n## Definition\n\nLet $X$ be a [[Banach space|06GZ956657]], let $K \\colon X \\to X$ be a bounded linear operator, typically a compact integral operator, and let $f \\in X$.\nConsider the [[equation of the second kind|8KJCN49V64]] $\\varphi - K\\varphi = f$.\nLet $X_1, X_2, \\ldots$ be finite-dimensional subspaces of $X$.\nFor each $n$ let $P_n \\colon X \\to X$ be a bounded linear [[projection|033DA1WE8Y]] with image $X_n$, that is $P_n^2 = P_n$ and $P_n(X) = X_n$.\nThe *projection method* defined by $(X_n, P_n)$ approximates $\\varphi$ by the solution $\\varphi_n \\in X_n$ of\n$$\\varphi_n - P_n K \\varphi_n = P_n f.$$\nIf $u_1, \\ldots, u_N$ is a basis of $X_n$ and $\\varphi_n = \\sum_j c_j u_j$, this equation is a system of $N$ linear equations for $c_1, \\ldots, c_N$.\n\n## Remarks\n\nTwo choices of $P_n$ are standard.\nIn the *collocation method*, $X_n \\subseteq C[a, b]$ and $P_n$ is interpolation at points $x_1, \\ldots, x_N$, so the equation is required to hold at these points.\nIn the *Galerkin method*, $X$ is a Hilbert space and $P_n$ is the [[orthogonal projection|0B8KZRT28M]] onto $X_n$.\nSo the residual is required to be orthogonal to $X_n$; see the [[Galerkin method|Y5B0PEBKE8]].\n\nExample: $X = L^2(0, 1)$, $(K\\varphi)(x) = \\int_0^1 x t \\, \\varphi(t) \\, dt$, $f(x) = x$, and $X_1$ spanned by $u_1(x) = x$ with $P_1$ orthogonal.\nWith $\\varphi_1 = c x$, one has $K\\varphi_1 = \\frac{c}{3} x$, so the equation is $c - \\frac{c}{3} = 1$ and $c = \\frac32$.\nSince $K$ maps into $X_1$ and $f \\in X_1$, this is the exact solution: $\\frac32 x - \\frac{x}{3} \\cdot \\frac32 = x$.\n\nNon-example: the Nyström method replaces the integral in $K$ by a quadrature rule and solves for the values of the unknown at the nodes.\nIt approximates the operator $K$ instead of projecting the unknown onto a fixed subspace, so it is not a projection method in this sense.\nIts analysis uses [[collectively compact operator approximation|AEZE6B2XXY]].\n\nThe convergence of projection methods for second-kind equations is described by the [[convergence theorem for second-kind equations|H6KS0E9MF5]].\n",
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