On Kakutani's characterization of the closed linear sublattices of $C(X)$ -- Revisited

In his paper [Concrete representation of abstract $(M)$-spaces. (A characterization of the space of continuous functions.), Ann. of Math., $42(2)$ ($1941$), $994$–$1024$.], S. Kakutani gave an interesting representation of the closed linear sublattices of the space of real-valued continuous functions on a compact Hausdorff space, which is determined by a set of algebraic relations. In this short note, we present a simple proof of this representation without using any profound lattice theory or functional analysis results, making this proof accessible even to undergraduate students.