1. The problem statement, all variables and given/known data Let E be a dense linear subspace of a normed vector space X, and let Y be a Banach space. Suppose T0 [itex]\in[/itex] £(E, Y) is a bounded linear operator from E to Y. Show that T0 can be extended to T[itex]\in[/itex] £(E, Y) (by continuity) without increasing its norm. 3. The attempt at a solution Someone kindly gave me a hint. I am trying to work out the details. The deadline is approaching. So I put the question here just in case. Thanks. For this particular problem you want to show that if (xn) converges to x then T0(xn) is a Cauchy sequence and then define f(x) as the limit of the sequence. Finally you need to show that the map is a well-defined bounded linear function.