A lower bound for the transpose forces a dense image of a ball

Given: 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.

From The transpose of a bounded operator, with its stated hypotheses: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. Let $T:X\to Y$ be bounded and linear between normed spaces. Its transpose, or Banach adjoint, is $T^{\ast }:Y^{\ast }⟶X^{\ast },(T^{\ast }g)(x)=g(Tx)\mathrm{.}$ The duals are def-dual-space-of-a-normed-space. Composition is bounded by lem-composition-operator-norm-inequality, so this has the displayed codomain. It is linear in $g$ over $\mathbb{K}$. No complex conjugation is inserted; a Hilbert adjoint uses a separate inner-product identification.

From The transpose is bounded with the same norm, with its stated hypotheses: Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$. For a bounded linear $T:X\to Y$ between normed spaces, $T^{\ast }:Y^{\ast }\to X^{\ast }$ is bounded linear and $\mathrm{\parallel }{T}^{\ast }\mathrm{\parallel }=\mathrm{\parallel }T\mathrm{\parallel }$.

From Strong separation of a closed and a compact convex set, with its stated hypotheses: Let $C,K\subseteq X$ be disjoint nonempty convex sets, where $C$ is closed and $K$ is compact. Then they are strongly separated by a nonzero functional in $X^{\ast }$.