Generalization of Hahn-Banach Theorem

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

:)

• Can you extend each coordinate without increasing the norm? Would that help? – GEdgar Mar 18 '14 at 0:52

If $T:Y\to\ell^\infty$ is linear and continuous there is one semi-norm $p$ on $X$ with $$\|T(y)\|_{\ell^\infty} \le p(y) \text{ for all y\in Y}.$$ As you said, you can apply Hahn-Banach to each coordinate of $T$ (always with the same dominating $p$).