Evaluation onto the dual of a reflexive dual subspace ($T:Z\twoheadrightarrow X^{\ast },(Tz)(x)=x(z)$)

If $Z$ is a Banach space and $X$ is a norm-closed reflexive linear subspace of $Z^{\ast }$, evaluation gives the displayed surjection. Restriction $R:Z^{\ast \ast }\to X^{\ast }$ is weak-star continuous and maps the closed unit ball onto the target ball by the Hahn-Banach theorem. The Goldstine theorem then makes $T(B_Z)$ weak-star dense in $B_{X^{\ast }}$, where $T=R{J}_Z$. Reflexivity of $X$ makes this target weak-star topology equal to its weak topology. Convexity and the Hahn-Banach separation theorem convert weak density of this ball image to norm density.

To pass from density to surjectivity, approximate a desired $f$ by $T{z}_1$ with $\Vert z_1\Vert \le \Vert f\Vert$ and residual at most $\Vert f\Vert /2$. Repeat on each residual. The chosen vectors have summable norms, with sum at most $2\Vert f\Vert$, and the residuals tend to zero geometrically. Completeness of $Z$ produces $z={\sum }_n{z}_n$ with $Tz=f$. Using any smaller residual ratio proves the infimum lifting norm equals $\Vert f\Vert$, although that infimum need not be attained. This argument proves onto-ness, rather than merely density of the evaluation functionals.