Let $X$ be a compact Hausdorff space and $K$ be a compact subspace of $X$. I am required to show that $C(X)/\{f:f|_K=0\}$ is isometrically isomorphism to $C(K)$. The norm used here the usual sup-norm.
I have shown that the quotient map induced by $\phi:f\mapsto f|_K$ is a surjective continuous map. The surjection follows from Tietze Extension Theorem. I am not sure how to proceed with the isometry part, although I have a feeling I have to use Urysohn's lemma somewhere.
Any help is highly appreciated.
Thanks
Best Answer
Hints:
Let $Z_K:=\{f:f|_K=0\}$ and $\alpha:C(X)/Z_K\to C(K)$, $f+Z_K\mapsto f|_K$ be the map of the question.
For showing the isometry part, let $f\in C(X)$ be given. We have to show that $\|f+Z_K\|=\|f_K\|$ holds.
First, show the inequality $\|f+Z_K\|\geq \|f|_K\|$ using the definition of the norms. For the other inequality, first find a function $g\in C(X)$ such that $g=f$ on $K$ but $\|g\|\leq \|f|_K\|$ (how can you construct such a function $g$?). Then the other inequality follows because of $f+Z_K=g+Z_K$.