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  "body": "\n## Statement\n\nLet $U$ and $V$ be real [[Hilbert spaces|HM2HHE3J51]].\nLet $a \\colon U \\times V \\to \\R$ be a [[bilinear form|0SQZMXB6GP]] with $\\abs{a(w, v)} \\le M \\norm{w}_U \\norm{v}_V$ for all $w \\in U$ and $v \\in V$, and let $f$ be in the [[continuous dual space|4A82S5FVRT]] $V'$.\nLet $u \\in U$ satisfy $a(u, v) = f(v)$ for all $v \\in V$.\nLet $U_h \\subseteq U$ (trial space) and $V_h \\subseteq V$ (test space) be finite-dimensional subspaces of the same dimension, with\n$$\\beta_h = \\inf_{w \\in U_h \\setminus \\set{0}} \\ \\sup_{v \\in V_h \\setminus \\set{0}} \\frac{a(w, v)}{\\norm{w}_U \\norm{v}_V} \u003e 0.$$\nThen there is a unique $u_h \\in U_h$ with $a(u_h, v) = f(v)$ for all $v \\in V_h$, and\n$$\\norm{u - u_h}_U \\le \\Big( 1 + \\frac{M}{\\beta_h} \\Big) \\inf_{w \\in U_h} \\norm{u - w}_U.$$\n\n## Remarks\n\nIn bases of $U_h$ and $V_h$ the discrete problem is a square linear system.\nThe condition $\\beta_h \u003e 0$ says that $a(w, v) = 0$ for all $v \\in V_h$ forces $w = 0$, so the system matrix is injective and therefore invertible.\nSubtracting the equations gives $a(u - u_h, v) = 0$ for all $v \\in V_h$.\nFor $w \\in U_h$ this gives $\\beta_h \\norm{u_h - w} \\le \\sup_{v} a(u_h - w, v) / \\norm{v} = \\sup_{v} a(u - w, v) / \\norm{v} \\le M \\norm{u - w}$.\nThe triangle inequality $\\norm{u - u_h} \\le \\norm{u - w} + \\norm{w - u_h}$ then gives the bound.\n\nThe case $U = V$ and $U_h = V_h$ is the [[Galerkin method|Y5B0PEBKE8]]; there, coercivity with constant $\\alpha$ gives $\\beta_h \\ge \\alpha$.\nExample: in the streamline upwind Petrov–Galerkin method for $-\\varepsilon u'' + u' = f$, each test function is a trial hat function plus a multiple of its derivative, which adds stabilisation along the flow.\n\nA positive $\\beta_h$ is needed: with $\\beta_h = 0$ the discrete system is singular, and with $\\beta_h \\to 0$ along a family of meshes the bound degenerates.\n\n## Explanations\n\n### L1 human\n\nIn the Galerkin method one searches and tests in the same space of functions.\nThe Petrov–Galerkin method allows two different spaces: one where the approximate solution lives, and one that is used to test the equation.\nThe theorem says that the method works, and gives a nearly best approximation, as long as the two spaces fit together well, as measured by the number $\\beta_h$.\n\n### L2 human\n\nThe theorem is the discrete half of the Babuška theory of problems that are well posed but not coercive.\nIt explains why different test spaces can stabilise convection-dominated problems and why some choices of spaces fail.\nIts constant $1 + M / \\beta_h$ can be improved to $M / \\beta_h$ in Hilbert spaces, a result of Xu and Zikatanov.\nFor saddle-point problems, the analogous condition is the [[Ladyzhenskaya–Babuška–Brezzi condition|SC5FV3GHV5]].\n",
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