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  "body": "\n## Definition\n\nLet $E$ be a real [[Banach space|06GZ956657]] with norm $\\norm{\\cdot}$, and let $\\le$ be a [[partial order|0QV2JS9RJ3]] on $E$.\nThe pair is a *Banach lattice* if the following hold.\n\n1. If $x \\le y$, then $x + z \\le y + z$ for all $z \\in E$.\n2. If $x \\le y$ and $t \\ge 0$ is real, then $t x \\le t y$.\n3. Every two elements $x, y$ have a least upper bound $x \\vee y$ in $E$.\n4. If $\\abs{x} \\le \\abs{y}$, then $\\norm{x} \\le \\norm{y}$, where $\\abs{x} = x \\vee (-x)$.\n\n## Remarks\n\nClauses 1 to 3 say that $E$ is a Riesz space, and clause 4 says that the norm is a lattice norm.\n\nExample: $C(K)$ for a compact Hausdorff space $K$, with $f \\le g$ when $f(t) \\le g(t)$ for all $t \\in K$.\nClauses 1 and 2 are clear, and $f \\vee g$ is the pointwise maximum, which is continuous.\nThen $\\abs{f}$ is the pointwise absolute value, and $\\abs{f} \\le \\abs{g}$ gives $\\sup \\abs{f} \\le \\sup \\abs{g}$.\n\nNon-example: $\\R^2$ with the coordinatewise order and the norm $\\norm{(a, b)} = \\abs{a} + \\abs{a - b}$.\nClauses 1 to 3 hold, and this is a norm.\nFor $x = (1, 0)$ and $y = (1, 1)$, one has $\\abs{x} \\le \\abs{y}$, but $\\norm{x} = 2 \u003e 1 = \\norm{y}$, so clause 4 fails.\n",
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