Extension of a bounded linear operator from a dense subspace

Let $D$ be a dense linear subspace of a normed space $X$, let $Y$ be a Banach space, and let $T:D\to Y$ be linear with $\Vert Tx\Vert \le C\Vert x\Vert$. Then $T$ has a unique bounded extension $\tilde{T}:X\to Y$, defined by $\tilde{T}x={\lim }_{n}T{x}_n$ for any sequence $x_n\in D$ converging to $x$, and $\Vert \tilde{T}\Vert \le C$.

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 ${\ell }^2$.