Isometric Isomorphism between Banach Spaces

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

• Try remembering how is usually defined the norm on quotients Oct 1, 2020 at 21:03
• Which of the two inequalities have you tried to show? How far did you get? Have you shown injectivity yet? Oct 1, 2020 at 21:56

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 $$f+Z_K=g+Z_K$$.