Bounded Linear Transformation to Banach Space has Unique Extension to Closure of Domain/Corollary 2

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

Let $\struct {X, \norm \cdot_X}$ be a normed vector space.

Let $\struct {Y, \norm \cdot_Y}$ be a Banach space. Let $D$ be an everywhere dense linear subspace of $X$.

Let $T_1 : X \to Y$ and $T_2 : X \to Y$ be bounded linear transformations with:


 * $T_1 x = T_2 x$ for all $x \in D$.

Then:


 * $T_1 = T_2$