Uniform Boundedness/Hahn-Banach Question Let $X=C(S)$ where $S$ is compact. Suppose $T\subset S$ is a closed
subset such that for every $g\in C(T),$ there is an $f\in C(S)$
such that: $f\mid_{T}=g$. Show that there exists a constant $C>0$
such that every $g\in C(T)$ can be continuously extended to $f\in C(S)$
such that: $\sup_{x\in S}\left|f(x)\right|\leq C\sup_{y\in T}\left|g(y)\right|$

So $A:C(S)/M \rightarrow C(T)$, then $A^{-1}:C(T)\rightarrow C(S)/M$,
apply the inverse mapping theorem, we obtain $\left\Vert A^{-1}g\right\Vert \leq\left\Vert A^{-1}\right\Vert \left\Vert g\right\Vert $,
then $\inf_{f'\in M}\left\Vert f+f^{'}\right\Vert \leq C\left\Vert g\right\Vert $.
We must show that there exists a function in $f+M$ whose restriction
is $g$ and it's norm is less than C$\left\Vert g\right\Vert $
But How to show the existence of such function?
