Generalized Stone approximation theorem for arbitrary algebras of functions (Q1594363)

Let \(C(X)\) denote the space of real-valued continuous functions on a compact Hausdorff space \(X.\) Let \(f\) be any real-valued nonlinear continuous function on the real line. Let \(E\) be a closed linear subspace of \(C(X)\) such that \(1\in E,\) functions from \(E\) separate points in \(X,\) and \(f\circ g\in E\) for any \(g\in E.\) Then the author shows that \(E=C(X).\) The author also discusses the algebraic counterpart of his result.