Image of Bounded Linear Transformation is Everywhere Dense iff Dual Operator is Injective

Theorem
Let $\GF \in \set {\R, \C}$.

Let $X$ and $Y$ be normed vector spaces over $\GF$.

Let $X^\ast$ and $Y^\ast$ be the normed dual spaces of $X$ and $Y$ respectively.

Let $T : X \to Y$ be a bounded linear transformation.

Let $T^\ast : Y^\ast \to X^\ast$ be the dual operator of $T$.

Then $T \sqbrk X$ is everywhere dense in $Y$ $T^\ast$ is injective.