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: $\sup_{x\in S}\left|f(x)\right|\leq C\sup_{y\in T}\left|g(y)\right|$

$C(S) \to C(T)$ is a surjective bounded linear map of Banach spaces (with sup norms), so there is a closed linear subspace $M \subset C(S)$ such that $C(S)/M \to T$ is bijective and bounded with the quotient norm. Inverse mapping theorem says that the inverse is a bounded linear map. The statement then follows.
By the way, how is this related to Banach-Steinhaus/Hahn-Banach?
