Compact Hausdorff space and its double dual: Gelfand Naimark. I am trying to prove the equivalence  of categories between compact Hausdorff spaces and unital $C^*$ algebras. 

The maps 
      \begin{align*}
 f: X \mapsto C(X) \\ 
 g: A \mapsto \hat{A}
 \end{align*}
       is a contravariant category equivalence between the category of compact Hausdorff spaces  and continuous maps and the category of commutative unital $C^*$ algebras and unital $*$-homomoprhisms. 

I know one direction $fg \simeq id:A \rightarrow C(\hat{A})$ is Gelfand Naimark. 
It then suffices to prove $gf \simeq id$, i.e. 
$$ X \mapsto \widehat{C(X)}, \quad x \mapsto \phi_x:f \mapsto f(x)$$
is a homeomorphism? 
 A: Yes. And the only nontrivial part is that the map $x\longmapsto \phi_x$ is surjective. Below is the argument that I know. 
Assume that $\varphi:C(X)\to\mathbb C$ is a character. We prove a few things


*

*if $\varphi(f)=0$, then there exists $x_f\in X$ such that $f(x_f)=0$. Indeed, if $f(x)\ne0$ for all $x$, then by compactness there exists $c>0$ with $|f(x)|\geq c$ for all $x$. Then $1/f\in C(X)$ and $1=\varphi(f/f)=\varphi(f)\varphi(1/f)$ and $\varphi(f)\ne0$. 

*for each $f\in C(X)$, there exists $x_f\in X$ with $\varphi(f)=f(t_x)$. Apply the above to the function $f-\varphi(f)$. 

*For each $f\in C(X)$, consider the set $T_f=\{x:\ f(x)=\varphi(f)\}$. By the above, $T_f\ne\varnothing$. As $T_f=f^{-1}(\{\varphi(f)\})$, it is compact. 

*Given $f_1,\ldots,f_n\in C(X)$, we have $\bigcap_{j=1}^n T_{f_j}\ne\varnothing$. Indeed, let 
$$
g=\sum_{j=1}^n|f_j-\varphi(f_j)|^2. 
$$
As $\varphi$ is linear and multiplicative,
$$
\varphi(g)=\sum_{j=1}^n \varphi[(f_j-\varphi(f_j))\overline{(f_j-\varphi(f_j))}]
=\sum_{j=1}^n \varphi[f_j-\varphi(f_j)]\varphi[\overline{f_j-\varphi(f_j)}]=0
$$
As we showed above, there exists $x\in X$ with $g(x)=0$. It follows that $f_j(x)=\varphi(f_j)$ for $j=1,\ldots,n$ so $x\in\bigcap_{j=1}^n T_{f_j}$. 

*The family $\{T_f\}_{f\in C(X)}$ has the finite intersection property. As $X$ is compact, it follows that $\bigcap_{f\in C(X)}T_f\ne\varnothing$. That is, there exists $x\in X$ such that $\varphi(f)=f(x)$ for all $f\in C(X)$. 
