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  "body": "\n## Definition\n\nLet $H$ be a separable [[Hilbert space|HM2HHE3J51]].\nA sequence $(x_n)_{n \\ge 1}$ in $H$ is a *Riesz sequence* if there are constants $0 \u003c A \\le B \u003c \\infty$ such that for every finite sequence of scalars $c_1, \\dots, c_N$,\n$$A \\sum_{n=1}^{N} \\abs{c_n}^2 \\le \\Big\\| \\sum_{n=1}^{N} c_n x_n \\Big\\|^2 \\le B \\sum_{n=1}^{N} \\abs{c_n}^2.$$\nA *Riesz basis* of $H$ is a Riesz sequence whose closed linear span is $H$.\n\n## Remarks\n\nEquivalently, $(x_n)$ is a Riesz basis if and only if $x_n = T e_n$ for an [[orthonormal basis|0S9GGTR0EX]] $(e_n)$ of $H$ and a bounded linear operator $T \\colon H \\to H$ with a bounded inverse.\nThen every $x \\in H$ has a unique expansion $x = \\sum_n c_n x_n$, the series converges unconditionally, and $A \\sum \\abs{c_n}^2 \\le \\norm{x}^2 \\le B \\sum \\abs{c_n}^2$.\nThe best constants are $A = \\norm{T^{-1}}^{-2}$ and $B = \\norm{T}^2$.\n\nExample: let $(e_n)$ be an orthonormal basis of $H$ and $x_n = e_n + \\frac12 e_{n+1}$.\nThen $x_n = T e_n$ with $T = I + \\frac12 S$, where $S$ is the shift $e_n \\mapsto e_{n+1}$, of norm $1$.\nSince $\\norm{\\frac12 S} \u003c 1$, the Neumann series shows that $T$ is invertible with bounded inverse, so $(x_n)$ is a Riesz basis.\n\nNon-example: $x_n = e_n / n$.\nIts closed span is $H$ and the upper bound holds with $B = 1$.\nThe lower bound fails: for $c = (0, \\dots, 0, 1)$ with the $1$ in place $n$, the middle term is $1/n^2$, which is not bounded below by a positive $A$.\n",
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