Bidual and canonical embedding

Remarks

For each $x$, the map $Jx$ is linear on $X^*$ and $\abs{(Jx)(f)} \le \norm{f}\,\norm{x}$, so $Jx \in X^{**}$ with $\norm{Jx} \le \norm{x}$. The Hahn–Banach theorem gives $f$ with $\norm{f} = 1$ and $f(x) = \norm{x}$, so in fact $J$ is a linear isometry. Example: for $X = \mathbb{K}^n$ with any norm, $J$ is an isometric isomorphism onto $X^{**}$, because $\dim X^{**} = \dim X$; compare the double dual in linear algebra. Example: for $X = c_0$ one has $X^* = \ell^1$ and $X^{**} = \ell^\infty$, and $J$ is the inclusion of $c_0$ in $\ell^\infty$, which is not onto. Non-example: the algebraic double dual, the space of all linear forms on $X^*$, is in general larger than $X^{**}$, because it does not require continuity. The image $J(X)$ is closed in $X^{**}$ exactly when $X$ is complete; in general the closure of $J(X)$ is a completion of $X$.