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: $\sup_{x\in S}\left|f(x)\right|\leq C\sup_{y\in T}\left|g(y)\right|$
Uniform Boundedness/Hahn-Banach Question
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functional-analysis
banach-spaces