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

This category contains pages concerning Image of Bounded Linear Transformation is Everywhere Dense iff Dual Operator is Injective:

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 $Y$ if and only if $T^\ast$ is injective.