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  "body": "\n## Statement\n\nLet $H$ be an infinite-dimensional real or complex [[Hilbert space|HM2HHE3J51]].\nLet $A$ be a [[self-adjoint operator|0NEGN5CRSC]] in $H$ that is bounded below and has compact resolvent.\nLet $\\lambda_1 \\le \\lambda_2 \\le \\lambda_3 \\le \\cdots$ be its [[eigenvalues|0BQAA2BK3Q]], listed with multiplicity.\nLet $Q$ be the closed quadratic form of $A$, with form domain $D(Q)$, so that $Q(u) = \\ip{Au}{u}$ for $u$ in the domain of $A$.\nFor $u \\in D(Q)$ with $u \\ne 0$, let $R(u) = Q(u) / \\norm{u}^2$ be the [[Rayleigh quotient|06WHP8SE98]].\nThen, for every $k \\ge 1$,\n$$\\lambda_k = \\min_{\\substack{V \\subseteq D(Q) \\\\ \\dim V = k}} \\; \\max_{u \\in V \\setminus \\set{0}} R(u) = \\max_{\\substack{W \\subseteq H \\\\ \\dim W \\le k - 1}} \\; \\inf_{\\substack{u \\in D(Q) \\setminus \\set{0} \\\\ u \\perp W}} R(u),$$\nwhere $V$ and $W$ run over [[linear subspaces|00YC1D652K]] of the given dimensions.\n\n## Remarks\n\nProof idea: let $(e_j)$ be an orthonormal basis of eigenvectors with $A e_j = \\lambda_j e_j$.\nOn the span of $e_1, \\dots, e_k$ the quotient $R$ is at most $\\lambda_k$.\nEvery $k$-dimensional subspace $V$ contains a nonzero vector orthogonal to $e_1, \\dots, e_{k-1}$, and there $R \\ge \\lambda_k$ by expanding in the basis.\nThe max-min form is proved in the same way, with the orthogonal complement of $W$.\n\nExample: for a bounded open set $\\Omega \\subseteq \\R^n$ and $A = -\\Delta$ with Dirichlet condition, $D(Q) = H^1_0(\\Omega)$ and $R(u) = \\int_\\Omega \\abs{\\nabla u}^2 / \\int_\\Omega u^2$.\nSo the [[Dirichlet eigenvalues|4VD8XXQ4P3]] are $\\lambda_k(\\Omega) = \\min_{\\dim V = k} \\max_{u \\in V \\setminus \\set{0}} R(u)$.\nSince $H^1_0(\\Omega_1) \\subseteq H^1_0(\\Omega_2)$ when $\\Omega_1 \\subseteq \\Omega_2$, this gives domain monotonicity: $\\lambda_k(\\Omega_2) \\le \\lambda_k(\\Omega_1)$.\n\nThe compact resolvent is used to have a basis of eigenvectors.\nWithout it the formulas still hold for the eigenvalues below the bottom of the essential spectrum, and otherwise the min-max values equal the bottom of the essential spectrum.\nIn finite dimension the result is the [[Courant–Fischer min-max theorem|CJK2HS2508]] for Hermitian matrices.\n\n## Explanations\n\n### L1 human\n\nThe eigenvalues of a symmetric operator can be found by optimisation alone.\nThe lowest one is the smallest value of the energy ratio $R$.\nThe $k$-th one is the smallest value one can reach for the worst vector of a $k$-dimensional family of trial functions.\n\n### L2 human\n\nThe theorem turns questions on eigenvalues into questions on trial functions.\nAny $k$-dimensional subspace of trial functions gives the upper bound $\\lambda_k \\le \\max_V R$, which is the basis of the Rayleigh–Ritz and finite element approximations of eigenvalues.\nIt gives domain monotonicity of Dirichlet eigenvalues, the comparison of Dirichlet and Neumann eigenvalues, and, with Weyl's counting argument, the asymptotic law of eigenvalues.\nFor the Laplacian, the first eigenvalue is the minimum of the Rayleigh quotient, which is the starting point of the [[Faber–Krahn inequality|E7TY2TPXX7]].\n",
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