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  "body": "\n## Definition\n\nLet $V$ be a real [[vector space|05EGHEQZAW]] and let $\\le$ be a [[partial order|011C29Q1S1]] on $V$.\nThe pair $(V, \\le)$ is an *ordered vector space* if for all $x, y, z \\in V$ and all real $\\lambda \\ge 0$:\n\n1. if $x \\le y$, then $x + z \\le y + z$;\n2. if $x \\le y$, then $\\lambda x \\le \\lambda y$.\n\n## Remarks\n\nExample: $\\R^n$ with the componentwise order is an ordered vector space.\nIf $x_i \\le y_i$ for all $i$, then $x_i + z_i \\le y_i + z_i$ and $\\lambda x_i \\le \\lambda y_i$ for $\\lambda \\ge 0$.\nNon-example: on $\\R$, let $x \\preceq y$ when $x = y$ or $(x, y) = (0, 1)$.\nThis is a partial order, but $0 \\preceq 1$ while $1 \\not\\preceq 2$, so clause 1 fails.\nThe set $\\set{x \\in V : 0 \\le x}$ is a [[positive cone|WG6BY4SWTV]], and the order is recovered from it by $x \\le y \\iff y - x \\ge 0$.\nSo giving an ordered vector space is the same as giving a real vector space with a positive cone.\n",
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