Extension of a bounded linear operator from a dense subspace
= Extension of a bounded linear operator from a dense subspace
Let $D$ be a <dense subset>[dense linear subspace] of a normed space $X$, let $Y$ be a <Banach space>, and let $T:D\to Y$ be linear with $\lVert Tx\rVert\leq C\lVert x\rVert$. Then $T$ has a unique bounded extension $\widetilde T:X\to Y$, defined by $\widetilde Tx=\lim_nTx_n$ for any sequence $x_n\in D$ converging to $x$, and $\lVert\widetilde T\rVert\leq C$.