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  "body": "\n## Statement\n\nLet $H$ be a real [[Hilbert space|HM2HHE3J51]] with norm $\\norm{\\cdot}$.\nLet $B \\colon H \\times H \\to \\R$ be a [[bilinear form|0SQZMXB6GP]] with these two properties:\n\n1. $B$ is bounded: there is $M \\ge 0$ with $\\abs{B(u, v)} \\le M \\norm{u} \\norm{v}$ for all $u, v \\in H$.\n2. $B$ is [[coercive|86KRHS6X22]]: there is $\\alpha \u003e 0$ with $B(u, u) \\ge \\alpha \\norm{u}^2$ for all $u \\in H$.\n\nThen for every bounded linear functional $f$ on $H$, an element of the [[continuous dual space|4A82S5FVRT]] $H'$, there is a unique $u \\in H$ such that\n$$B(u, v) = f(v) \\quad \\text{for all } v \\in H.$$\nMoreover $\\norm{u} \\le \\norm{f}_{H'} / \\alpha$.\n\n## Remarks\n\nThe form $B$ need not be symmetric.\nBy the [[Riesz representation theorem|TTC40N49M6]], there are a bounded linear operator $A$ on $H$ with $B(u, v) = \\ip{Au}{v}$ and an element $w$ with $f(v) = \\ip{w}{v}$.\nCoercivity gives $\\alpha \\norm{u} \\le \\norm{Au}$, so $A$ is injective with closed range.\nIf $v$ is orthogonal to the range, then $B(v, v) = \\ip{Av}{v} = 0$, so $v = 0$; hence $A$ is onto and $u = A^{-1} w$.\nThe bound follows from $\\alpha \\norm{u}^2 \\le B(u, u) = f(u) \\le \\norm{f}_{H'} \\norm{u}$.\nWhen $B$ is symmetric, the theorem is the Riesz representation theorem for the inner product $B$.\n\nExample: let $U \\subset \\R^n$ be a bounded open set and $g \\in L^2(U)$.\nOn $H = H^1_0(U)$, the [[Sobolev space|M4CEZ5K9HP]], take $B(u, v) = \\int_U \\nabla u \\cdot \\nabla v \\, dx$ and $f(v) = \\int_U g v \\, dx$.\nThe [[Poincaré inequality|JS2CCR3RNS]] shows that $B$ is coercive, so there is a unique [[weak solution|SJZX81A783]] $u \\in H^1_0(U)$ of $-\\Delta u = g$ with $u = 0$ on $\\partial U$.\n\nCoercivity cannot be dropped.\nOn $H = \\ell^2$, let $S$ be the right shift and $B(u, v) = \\ip{Su}{v}$, which is bounded.\nFor $f(v) = \\ip{e_1}{v}$, a solution would satisfy $S u = e_1$, which is impossible since the first coordinate of $Su$ is $0$.\n\n## Explanations\n\n### L1 human\n\nMany boundary value problems can be written as \"find $u$ such that $B(u, v) = f(v)$ for every test vector $v$\".\nThe theorem says that this problem has exactly one solution as soon as $B$ is bounded and positive in a strong sense.\nIt also says that the solution depends continuously on the data $f$.\n\n### L2 human\n\nThis is the standard existence theorem for weak solutions of linear elliptic equations in divergence form.\nOne checks boundedness from bounds on the coefficients and coercivity from uniform ellipticity together with the Poincaré inequality, possibly after adding a multiple of $u$ to the equation.\nThe proof is also the basis of finite element error analysis.\nRestricting the problem to a finite-dimensional subspace gives the Galerkin approximation, and the constants $M$ and $\\alpha$ control its error.\nThe Babuška version replaces coercivity by an inf-sup condition and allows different trial and test spaces.\n",
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