  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.
