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

TL;DR

This paper revisits Kakutani's classical characterization of closed linear sublattices of continuous functions on compact spaces, providing a simplified proof accessible to undergraduates.

Contribution

It offers a straightforward proof of Kakutani's representation theorem without advanced lattice theory or functional analysis.

Findings

Simplified proof of Kakutani's theorem

Accessible to undergraduate students

Clarifies algebraic relations defining sublattices

Abstract

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.