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  "body": "\n## Definition\n\nLet $X$ be a [[normed space|03J5RAKB30]] and let $\\Sigma_0 \\subseteq \\Sigma_1 \\subseteq \\Sigma_2 \\subseteq \\cdots$ be subsets of $X$, which in general are not linear subspaces.\nUsually there is a constant $c$ with $\\Sigma_n + \\Sigma_n \\subseteq \\Sigma_{cn}$.\n*Nonlinear approximation* is the approximation of elements $f \\in X$ by elements of $\\Sigma_n$, measured by the [[error of best approximation|YDN0T4MEX0]]\n$$\\sigma_n(f) = \\inf_{s \\in \\Sigma_n} \\norm{f - s}_X.$$\nThe main example is $n$-term approximation from a dictionary $\\mathcal{D} \\subseteq X$, with $\\Sigma_n = \\set{\\sum_{k=1}^{n} c_k g_k : c_k \\in \\R,\\ g_k \\in \\mathcal{D}}$.\n\n## Remarks\n\nExample: let $X$ be a separable Hilbert space with [[orthonormal basis|0S9GGTR0EX]] $(e_j)$, and let $\\Sigma_n$ be the vectors with at most $n$ nonzero coordinates.\nThe set $\\Sigma_1$ is not a subspace: $e_1, e_2 \\in \\Sigma_1$ but $e_1 + e_2 \\notin \\Sigma_1$.\nFor $f = \\sum_j c_j e_j$, the best element of $\\Sigma_n$ keeps the $n$ coefficients of largest absolute value, so $\\sigma_n(f)^2$ is the sum of $\\abs{c_j}^2$ over all other $j$.\n\nNon-example: $\\Sigma_n$ the polynomials of degree at most $n$ in $C[0, 1]$ is a linear subspace, so approximation from it is linear approximation.\n",
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