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  "body": "\n## Summary\n\nAn ordered vector space is a real [[vector space|000AYPA9AJ]] with a compatible [[partial order|0QV2JS9RJ3]].\nThe order is preserved by adding a vector and by multiplying by a nonnegative scalar.\nSuch an order is the same as a [[positive cone|Z344RY5XX1]], the set of vectors that are at least $0$.\nSpaces of real functions, of sequences and of self-adjoint operators carry natural orders of this kind.\nOrdered vector spaces are the setting of positive linear maps, and [[Riesz spaces|99709K2NJM]] are the case where the order is a lattice.\n\n## Explanations\n\n### L1 human\n\nIn $\\R^2$ say that $(a, b) \\le (c, d)$ when $a \\le c$ and $b \\le d$.\nMoving both points by the same vector keeps this relation, and so does stretching both by a positive factor.\nSome pairs of vectors are not comparable, such as $(1, 0)$ and $(0, 1)$.\nAn ordered vector space is any real vector space with an order that behaves like this.\n",
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