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  "body": "\n## Summary\n\nA Banach lattice is a [[Banach space|06GZ956657]] with an order that is compatible with the vector operations and with the norm.\nEvery two elements have a largest lower bound and a least upper bound, and a smaller absolute value gives a smaller norm.\nThe spaces $L^p$, $C(K)$ and $\\ell^p$ with the pointwise order are the main examples.\nBanach lattices are the setting for positive operators, which appear in probability, in partial differential equations and in economics.\n\n## Explanations\n\n### L1 human\n\nIn many function spaces you can compare two functions point by point and take their pointwise maximum.\nA Banach lattice keeps exactly this structure, in an abstract form, together with a complete norm.\nThe rule linking order and norm says that if $\\abs{f} \\le \\abs{g}$ everywhere, then $f$ is not longer than $g$.\nFor example, in $C([0,1])$ the function $\\max(\\sin, \\cos)$ is again continuous, and its norm is at most $1$ because both $\\abs{\\sin}$ and $\\abs{\\cos}$ are at most $1$.\n",
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