Banach-Stone Theorem In Fleming and Jamison's book, Banach first proved a lemma which used directional derivative to identity peak point of functions. Then he used the lemma in the proof of Banach-Stone theorem. After several years, Stone extended the result by loosening the condition compact metric space to compact Hausdorff space. But I can't seem to find any proof by Stone on how he proved the theorem with just only Hausdorff space. If anyone came across the original proof, can share the link here? or anyone still remember the idea, can share it?

Here is the refrence to the original article, see theorem 83.

The main idea of the proof is the following. Let $Z$ be a compact Hausdorff space. For a given $p\in Z$ denote $M_Z(p)=\\{f\in C(Z):|f(p)|=\Vert f\Vert\\}$. Also denote $\mathcal{M}_Z=\\{M_Z(p):p\in Z\\}$. Given $M\in\mathcal{M}_Z$ we can always recover its point $p\in Z$.

Once we have a surjective isometry $V:C(X)\to C(Y)$ we can establish a bijective correspondence between sets $\mathcal{M}_Y$ and $\mathcal{M}_X$. This correspondence gives rise to the bijection $\rho:Y\to X:q\mapsto p$ which turns out to be a homemorphism.