On “Lipschitz” subspaces of the space of continuous functions

Abstract

A theorem of Grothendieck states that every closed subspace of the Banach space \(L^p(\mu)\), where \(\mu\) is a finite measure on a locally compact topological space, \(p \ge 1\), consisting of essentially bounded functions must have finite dimension. An analog of this result is proved concerning subspaces of the space of continuous functions on a compact metric space consisting of functions satisfying different Lipschitz-type conditions.