Bidual of a normed space
= Bidual of a normed space
{title2=$X^{**}$}
= Bidual space
{synonym}
The bidual of a <normed vector space> $X$ is the continuous dual $X^{**}=(X^*)^*$. Evaluation defines the <canonical embedding into the bidual> $J_X(x)(f)=f(x)$. The <norm> inequality gives $\|J_Xx\|\le\|x\|$, and a norming functional from the <Hahn-Banach theorem> gives equality. Thus $J_X$ is isometric, and its image is dense in its closure, a <completion of a normed space>.