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  "body": "\n## Statement\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nLet $V^{**} = (V^*)^*$ be the dual of the [[dual space|07JBSZE0V4]].\nThe *evaluation map* $\\operatorname{ev} \\colon V \\to V^{**}$ is defined by $\\operatorname{ev}(v)(f) = f(v)$ for $v \\in V$ and $f \\in V^*$.\nThis map is linear and injective.\nIf $V$ is finite-dimensional, then $\\operatorname{ev}$ is an isomorphism.\n\n## Remarks\n\nThe evaluation map does not depend on a choice of basis, so it is called *canonical* or *natural*.\nIn finite dimensions, it is an isomorphism because it is injective and $\\dim V^{**} = \\dim V^* = \\dim V$.\nFor infinite-dimensional $V$, the evaluation map is not surjective.\n\n## Explanations\n\n### L1 human\n\nA vector $v$ can act on linear forms: it sends each form $f$ to the number $f(v)$.\nThis turns each vector into an element of the double dual.\nIn finite dimensions, every element of the double dual comes from exactly one vector in this way.\n",
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