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  "body": "\n## Definition\n\nLet $V$ be a real vector space with a partial order $\\le$ such that $(V, \\le)$ is an [[ordered vector space|X32V6Y5PYW]].\nThe space $V$ is a *Riesz space*, or *vector lattice*, if $(V, \\le)$ is a [[lattice|WH4SFAACAS]].\nThat is, any two elements $x, y \\in V$ have a least upper bound $x \\vee y$ and a greatest lower bound $x \\wedge y$.\n\n## Remarks\n\nExample: $\\R^n$ with the componentwise order is a Riesz space, with $x \\vee y$ the componentwise maximum.\nExample: the space $C([0, 1])$ of [[continuous functions|0P27W9RW0G]] with the pointwise order is a Riesz space, since $\\max(f, g)$ and $\\min(f, g)$ are continuous.\nNon-example: on $\\R^2$, let $(a, b) \\le (c, d)$ when $a \\le c$ and $b = d$.\nThis makes $\\R^2$ an ordered vector space, but $(0, 0)$ and $(0, 1)$ have no common upper bound, so it is not a lattice.\nIn a Riesz space one writes $x^+ = x \\vee 0$, $x^- = (-x) \\vee 0$ and $\\abs{x} = x \\vee (-x)$, and then $x = x^+ - x^-$ and $\\abs{x} = x^+ + x^-$.\nA Riesz space with a complete norm satisfying $\\abs{x} \\le \\abs{y} \\implies \\norm{x} \\le \\norm{y}$ is a [[Banach lattice|G4CA22W2XX]].\n",
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