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Let $X$ be a compact metric space and let $F$ be a closed subset of $X$. Assume that there exists a bounded extension operator $T:C(F) \rightarrow C(X)$, i.e., $T \in B(C(F),C(X))$ and for all $g\in C(F)$, $T(g)|_{F} =g$.

How to prove that there is a subspace $W \subset C(X)$ so that $C(X)$ is isomorphic (as a vector space) to $W \bigoplus Z$, where $Z=\{f \in C(X) : f|_F =0 \}$?

Equip $W \bigoplus Z$ with the norm $\||(w,z)|\| = \|w\|+\|z\|$. The above isomorphism is an isomorphism of Banach spaces?

Let $X=[0,1]$ and let $F$ be a closed subset of $X$. How to prove there is a bounded extension operator $T$ from $C(F)$ to $C([0,1])$?

Thank you so much for your help.

1 Answers 1