Equibounded pointwise approximation implies the uniform one

Abstract Given any compact Hausdorff space X , we present a simple proof that a continuous function $$g\in \mathcal {C}(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> can be uniformly approximated on X by elements of some linear subspace $$\mathcal {L}\subset \mathcal {C}(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">