I have a question from Folland's book "Real Analysis" (p. 142. ex. 71. c)
Let $X$ be compact and Hausdorff space and $M$ be the set of all nonzero algebra homomorphisms from $C(X,\mathbb{R})$ to $\mathbb{R}$, where $C(X,\mathbb{R})$ is set of continuous functions from $X$ to the real line. Each $x$ in $X$ defines an element $y$ of $M$, by $y(f) = f(x)$.
The question is that prove that the map $x \mapsto y$ is a bijection from $X$ to $M$. Maybe its' easy to prove that its injective, but I can`t prove surjectivity. Also in same problem I proved that if $Q$ is the proper ideal in $C(X,\mathbb{R})$, then there exists $z$ in $X$, such that $f(z)=0$ for all $f$ in $Q$. But I am not sure that this fact can used in my question.