Equibounded pointwise approximation implies the uniform one

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