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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nThe *dual space* of $V$ is the vector space\n$$V^* = \\Hom_F(V, F)$$\nof all [[linear forms|07YQ1C78G0]] on $V$.\nIts operations are pointwise: $(f + g)(v) = f(v) + g(v)$ and $(af)(v) = a f(v)$.\n\n## Remarks\n\nIt is a special case of the [[space of linear maps|0X9MBPP2Z3]].\nIf $\\dim V = n$ is finite, then $\\dim V^* = n$, see [[the dimension of the dual space|00889YD15K]].\nThe elements of $F^n$ act on column vectors by the dot product, and this identifies $(F^n)^*$ with $F^n$.\nFor infinite-dimensional $V$, the space $V^*$ is not isomorphic to $V$.\n\n## Explanations\n\n### L1 human\n\nThe dual space collects all the ways to measure a vector by one linear number.\nOn $\\R^3$, each linear form is a dot product with a fixed vector.\nSo, in finite dimensions, the dual space looks like the original space, but its elements play a different role.\n",
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