TITLE: extending a bounded linear operator
QUESTION [10 upvotes]: So I have a homework question which I have no idea how to start.
Let $E_0$ be a dense subspace of the normed space $E$. Let $T_0:E_0 \rightarrow F$ be a bounded linear operator into the Banach space $F$.
(i) Show that $T_0$ can be uniquely extended to a bounded linear operator $T:E \rightarrow F$.
(ii) Prove that $\|T\| = \|T_0\|$.
Any hints would be much appreciated!

REPLY [2 votes]: General notions in this area are the concept of closable operator and the closed graph theorem. This matter is discussed in Kato's "Perturbation Theory of Linear Operators". Here the boundedness simplifies matters. See Problem 5.17 on p 166 (second corrected printing of second edition) which states that every bounded operator is closable. In the present case this means that $T_0$ indeed extends in a unique way to all of $E$.