Banach-Stone Theorem

Solution 1:

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.

Solution 2:

In here, the following article of Stone (1937) is referred:

Stone, M.H., Applications of the theory of boolean rings to General Topology, Trans. Amer. Math. Soc., 41 (1937), 375 – 481.