Bidual of a normed space ($X^{\ast \ast }$)

The bidual of a normed vector space $X$ is $X^{\ast \ast }=(X^{\ast }{)}^{\ast }$, 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 bidualThe dual norm gives $\Vert J_{X}x\Vert \le \Vert x\Vert$. For $x\ne 0$, the Hahn-Banach theorem supplies a norm-one linear functional with $|f(x)|=\Vert x\Vert$, 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 $X^{\ast \ast }$ 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.