$\newcommand{\M}{\mathcal{M}}\newcommand{\abs}[1]{\lvert #1 \rvert}\newcommand{\blank}{{-}}\newcommand{\from}{\colon}\newcommand{\IRpos}{\mathbb{R}_{\ge 0}}$For a commutative Banach ring $A$, e.g. a Banach Algebra over $\mathbb{R}$, let me write $\M(A)$ for the set of all multiplicative (for me, this includes $\abs{1} = 1$) seminorms $\abs{\blank} \from A \to \IRpos$, which are bounded by the given norm on $A$. The natural topology on this set is the coarsest topology such that for every $f \in A$ the function $\M(A) \to \IRpos$, $\abs{-} \mapsto \abs{f}$ is continuous. One can show in this generality that the space $\M(A)$ is a compact Hausdorff space (this is Theorem 1.2.1 in the monograph by Berkovich) and it is commonly referred to as the Berkovich spectrum of $A$. It was systemetically studied (though mainly in the case of non-archimedean Banach algebras) in the book Spectral Theory and Analytic Geometry over Non-Archimedean Fields by V. G. Berkovich.
Terry Tao has shown above that if $X$ is a compact Hausdorff Space, then the natural map $X \to \M(C(X))$ is a bijection. Equipping the right hand side with the above topology, it is continuous by construction and hence a homeomorphism since both sides are compact Hausdorff.