= Bidual evaluation map
{title2=$J(x)(T)=T(x)$}
For an algebraic <dual space> $X^*=\operatorname{Hom}_F(X,F)$, the <linear map> $J:X\to X^{**}$ sends $x$ to evaluation at $x$. It is <injective> because <linear functionals> separate nonzero vectors: extend any nonzero vector to a <basis> and use its coordinate functional. If $X$ has a finite <basis> $e_i$ with <dual basis> $e_i^*$, every $B\in X^{**}$ equals $J(\sum_iB(e_i^*)e_i)$, so $J$ is an <isomorphism>. For infinite-dimensional spaces, <surjection> does not follow from this finite sum argument.
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