Failure of dense-subspace extension into an incomplete codomain
= Failure of dense-subspace extension into an incomplete codomain
Let $c_{00}$ have the $\ell^2$ norm, take $D=c_{00}\subset X=\ell^2$, and let $T:D\to c_{00}$ be the identity. It is bounded but has no continuous extension $\ell^2\to c_{00}$, because continuity and density would force the extension to be the identity on every element of $\ell^2$.