Let $Y$ be a subspace of a locally convex topological vector space $X$. Suppose $T:Y\longrightarrow l^\infty$ is a continuous linear operator. Prove that $T$ can be extended to a continuous linear operator $\tilde T:X\longrightarrow l^\infty$ such that $\tilde T|_Y=T$.
We can extend each coordinate linear functional to a continuous linear functional on $X$. But I do not know how to prove the series is still in $l^\infty$. Note that the question gives locally convex topological vector spaces but not normed space. Hence I am not able to extend the coordinate functionals preserving their norms, which are not well-defined.
How to prove? Thank you a lot