Smooth approximations preserving asymptotic Lipschitz bounds

Abstract The goal of this note is to prove that every real-valued Lipschitz function on a Banach space can be pointwise approximated on a given $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> -compact set by smooth cylindrical functions whose asymptotic Lipschitz constants are controlled. This result has applications in the study of metric Sobolev and BV spaces: it implies that smooth cylindrical functions are dense in energy in these kinds of functional spaces defined over any weighted Banach space.