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  "body": "\n## Definition\n\nLet $\\ell^\\infty$ be the vector space of bounded real sequences $x = (x_n)_{n \\ge 1}$, as in the [[spaces ℓ^p|08P6T2WH6R]].\nLet $S \\colon \\ell^\\infty \\to \\ell^\\infty$ be the shift, $S(x_1, x_2, x_3, \\dots) = (x_2, x_3, \\dots)$, and let $\\mathbf{1} = (1, 1, 1, \\dots)$.\nA *Banach limit* is a linear map $\\varphi \\colon \\ell^\\infty \\to \\R$ such that:\n\n1. $\\varphi(x) \\ge 0$ for every $x$ with $x_n \\ge 0$ for all $n$.\n2. $\\varphi(Sx) = \\varphi(x)$ for every $x \\in \\ell^\\infty$.\n3. $\\varphi(\\mathbf{1}) = 1$.\n\n## Remarks\n\nEvery Banach limit satisfies $\\liminf_n x_n \\le \\varphi(x) \\le \\limsup_n x_n$.\nCheck of the left inequality: if $x_n \\ge c$ for all $n \\ge N$, then $S^N x - c \\mathbf{1}$ has nonnegative terms, so $\\varphi(x) = \\varphi(S^N x) \\ge c$.\nHence $\\varphi(x) = \\lim_n x_n$ for every convergent $x$, and $\\abs{\\varphi(x)} \\le \\sup_n \\abs{x_n}$, so $\\varphi$ is continuous of norm $1$.\nBanach limits exist by the [[Hahn–Banach theorem|QB5AE3VYZZ]], applied to the sublinear functional $p(x) = \\limsup_{m} \\frac{1}{m} \\sum_{k=1}^{m} x_k$ or a similar one.\n\nNon-example: $\\varphi(x) = \\limsup_n x_n$ satisfies clauses 1, 2 and 3, but it is not linear.\nFor $x = (1, 0, 1, 0, \\dots)$ and $y = Sx$ one has $\\varphi(x) + \\varphi(y) = 2$ but $\\varphi(x + y) = \\varphi(\\mathbf{1}) = 1$.\n\nNon-example: $\\varphi(x) = x_1$ is linear and positive with $\\varphi(\\mathbf{1}) = 1$, but it fails clause 2.\n",
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