Let $X_1\subset X$ be a subspace of $X$, an Normed linear space, and $\overline{X_1}$ the closure of $X_1$ in $X$. Let $T: X_1\to Y$ bounded lienar transformation, i.e. $|T(x)|\leq C|x|$ for some $C$ and all $x \in X_1$. Prove that $T$ has unique continuous extension to $\overline{X_1}$ and that this extension is linear and that is bounded.
I think this:
Let $\overline{T}:\overline{X_1}\to Y$ such that $\overline{T}(x)=T(x)$ for all $x\in X_1$, and $\overline{T}(x)=\lim_n T(x_n)$ with $\lim_n x_n=x$ for some $(x_n)$ sequence in $X_1$ and $x\in \overline{X_1}\setminus X_1$
Would this candidate be an extension?
How prove uniqueness?