= Double dual
{title2=$V^{**}=\operatorname{Hom}_F(V^*,F)$}
= Algebraic double dual
{synonym}
The algebraic double dual of a <vector space> is the <dual space> of its <algebraic dual>. An element assigns a <scalar> linearly to each <linear functional> on the original space. The <bidual evaluation map> sends $v$ to the rule $\ell\mapsto\ell(v)$. It is a canonical <linear isomorphism> in finite <dimension>, unlike an identification of $V$ with $V^*$, which generally requires an additional choice. For spaces with topology, using the <continuous dual space> instead defines the topological bidual; that is a distinct construction in infinite <dimension>.
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