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

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

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

Let $\map D {T_0}$ be a linear subspace of $X$.

Let $\struct {Y, \norm \cdot_Y}$ be a Banach space over $\Bbb F$.

Let $T_0 : \map D {T_0} \to Y$ be a bounded linear transformation.

Then there exists a unique bounded linear transformation $T : \map D T \to Y$ extending $T_0$ to $\map D T = \paren {\map D {T_0} }^-$.

Also see

 * Uniformly Continuous Function to Complete Metric Space has Unique Continuous Extension to Closure of Domain gives a more general result that is admissable here, but this special case is interesting enough to justify individual treatment.