Given any compact Hausdorff space X, we present a simple proof that a continuous function \(g\in \mathcal {C}(X)\) can be uniformly approximated on X by elements of some linear subspace \(\mathcal {L}\subset \mathcal {C}(X)\) , if and only if g can be pointwise approximated on X by some equibounded sequence in \(\mathcal {L}\) . Moreover, given any compactum \(K\subset \mathbb {C}\) , we also show that every \(f\in \mathcal {C}(K)\) can be uniformly approximated by rational functions (without poles on K), if and only if the complex conjugate \(w\mapsto \overline{w}\) can be pointwise approximated by functions holomorphic on K (no equibounded hypothesis required).
