= Evaluation onto the dual of a reflexive dual subspace
{title2=$T:Z\twoheadrightarrow X^*,\quad(Tz)(x)=x(z)$}
If $Z$ is a <Banach space> and $X$ is a norm-closed reflexive <linear subspace> of $Z^*$, evaluation gives the displayed surjection. Restriction $R:Z^{**}\to X^*$ is <weak-star topology>[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 $B_{X^*}$, 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 $Tz_1$ with $\|z_1\|\leq\|f\|$ and residual at most $\|f\|/2$. Repeat on each residual. The chosen <vectors> have summable <norms>, with sum at most $2\|f\|$, and the residuals tend to zero geometrically. Completeness of $Z$ produces $z=\sum_nz_n$ with $Tz=f$. Using any smaller residual ratio proves the infimum lifting <norm> equals $\|f\|$, although that infimum need not be attained. This argument proves onto-ness, rather than merely density of the evaluation functionals.
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