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"> <mml:mrow> <mml:mi>L</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> , if and only if g can be pointwise approximated on X by some equibounded sequence in $$\mathcal {L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> . Moreover, given any compactum $$K\subset \mathbb {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> , we also show that every $$f\in \mathcal {C}(K)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> can be uniformly approximated by rational functions (without poles on K ), if and only if the complex conjugate $$w\mapsto \overline{w}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo>↦</mml:mo> <mml:mover> <mml:mi>w</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> can be pointwise approximated by functions holomorphic on K (no equibounded hypothesis required).