= Bidual of a normed space
{title2=$X^{**}$}
= Bidual space
{synonym}
The bidual of a <normed vector space> $X$ is $X^{**}=(X^*)^*$, where both stars mean the <continuous dual space>. Its elements are bounded <linear functionals> on the space of bounded <linear functionals> on $X$. Evaluation gives the <canonical embedding into the bidual>
$$
J_X:X\longrightarrow X^{**},\qquad J_X(x)(f)=f(x).
$$
The dual <norm> gives $\|J_Xx\|\le\|x\|$. For $x\ne0$, the <Hahn-Banach theorem> supplies a norm-one <linear functional> with $|f(x)|=\|x\|$, proving equality. Thus $J_X$ is an <isometry>, even if $X$ is incomplete.
The bidual is a <Banach space>, because a <continuous dual space> is complete. Therefore the closure of $J_X(X)$ in $X^{**}$ is a <completion of a normed space> $X$: it is complete and contains a dense isometric copy of $X$. This explains the role of the closure, rather than merely saying that an image is dense in its own closure.
If $X$ is already a <Banach space>, $J_X(X)$ is closed. It equals the whole bidual precisely when $X$ is a <reflexive Banach space>. Completeness and reflexivity are different requirements; constructing the bidual does not by itself establish either that the original space was complete or that its canonical embedding is surjective.
Back to article page
