Kernel-range identities and the weak-star closure of the transpose range

Given: The Axiom of Choice, The spaces, maps, scalar field, and hypotheses in the statement above. All duals consist of linear functionals over the ambient field; evaluation has no conjugation.

AC is used through the preannihilator identity in [F1] at step 1.1, which separates nonzero values of $T$ by continuous functionals.

From Elementary kernel and range annihilator identities, with its stated hypotheses: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For a bounded linear $T:X\to Y$ between normed spaces, $(\mathrm{ran}T{)}^{\perp }=\ker T^{\ast },{}^{\perp }(\mathrm{ran}{T}^{\ast })=\ker T,{\overline{\mathrm{ran}T}}^{\mathrm{\parallel }\cdot \mathrm{\parallel }}={}^{\perp }(\ker T^{\ast })\mathrm{.}$ The closure in the last identity is in $Y$.

From The weak-star topology from finite evaluations, with its stated hypotheses: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For a normed $X$ with continuous dual $X^{\ast }$ from def-dual-space-of-a-normed-space, the weak-star topology $\sigma (X^{\ast },X)$ is the initial topology of all evaluations $f\mapsto f(x)$ into $\mathbb{K}$ with its usual topology. At $f_0$, a neighbourhood basis consists of $U(f_0;x_1,\dots ,x_n;\epsilon )=\{f\in X^{\ast }:\mathrm{\mid }(f-f_0)(x_j)\mathrm{\mid }<\epsilon (1\le j\le n)\},$ where $n$ is finite and $\epsilon >0$. For $n=0$ the set is all of $X^{\ast }$. Finite intersections of inverse images of scalar open sets form the initial-topology basis; at the given point, finitely many disks can be refined using their smallest positive radius. Weak-star closure means closure in this topology, not merely sequential closure.

From Double annihilators give norm and weak-star closures, with its stated hypotheses: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For a normed $X$ and a linear subspace $N\le X^{\ast }$, $({}^{\perp }N{)}^{\perp }={\bar{N}}^{\sigma (X^{\ast },X)}\mathrm{.}$ Consequently $N$ is weak-star closed if and only if $N=({}^{\perp }N{)}^{\perp }$, and weak-star dense in $X^{\ast }$ if and only if ${}^{\perp }N=\{0\}$. For a linear subspace $M\le X$, the primal formula is ${}^{\perp }(M^{\perp })={\bar{M}}^{\mathrm{\parallel }\cdot \mathrm{\parallel }}$.