Let $X$ and $Y$ be two Banach spaces over $\mathbb{C}$
Let $V$ be a dense subspace of $X$
Let $T : V \to Y$ be a bounded linear operator
We know that $T$ can be uniquely extended to a bounded linear transformation $S : X \to Y$ such that $\left\| T \right\| = \left\| S \right\|$
I would like to know if is it true that
$$ T \text{ is injective } \Longrightarrow S \text{ is injective } $$
thanks.