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  "body": "\n## Summary\n\nA Banach limit is a positive, shift-invariant linear functional on the space $\\ell^\\infty$ of bounded real sequences.\nIt gives the value $1$ to the constant sequence $1$.\nIt extends the ordinary limit from convergent sequences to all bounded sequences.\nBanach limits exist by the [[Hahn–Banach theorem|QB5AE3VYZZ]], but none can be written down explicitly, and there are many of them.\nThey define [[almost convergence|8HTHB1QPZY]] and are a basic example of an invariant mean.\n\n## Explanations\n\n### L1 human\n\nThe sequence $1, 0, 1, 0, \\dots$ has no limit, but it is natural to say that its \"average limit\" is $1/2$.\nA Banach limit is a rule that assigns such a value to every bounded sequence.\nIt must be linear, it must give a nonnegative value to a nonnegative sequence, and it must not change when you drop the first term.\nThese rules force the value $1/2$ for $1, 0, 1, 0, \\dots$ and the usual limit for every convergent sequence.\n",
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