Past exam of the functional analysis course of the University of Oxford / 2011 / 2 / iv

The domain carries $\sigma (Z^{\ast \ast },Z^{\ast })$, and the target carries $\sigma (X^{\ast },X)$. For each $x\in X\subseteq Z^{\ast }$,which is one of the defining weak-star continuous evaluations on $Z^{\ast \ast }$. Thus $R$ is weak-star continuous.

Restriction cannot increase a functional's norm, so $R(B_{Z^{\ast \ast }})\subseteq B_{X^{\ast }}$. Conversely, any $f\in B_{X^{\ast }}$ is a bounded linear functional on the subspace $X$ of $Z^{\ast }$. The Hahn-Banach theorem extends it to a functional $\theta \in (Z^{\ast }{)}^{\ast }=Z^{\ast \ast }$ with the same norm. Then $R\theta =f$ and $\Vert \theta \Vert \le 1$. ConsequentlyThis proves exact surjectivity on closed balls, not just density of the image.