Abstract
We show among other things that if B is a linear space of continuous real-valued functions vanishing at infinity on a locally compact Hausdorff space X, for which there is a continuous function h defined in a neighbourhood of 0 in the real line which is non-affine in every neighbourhood of 0 and satisfies |h(t)|⩽k|t| for all t, such that hb is in B whenever b is in B and the composite function is defined, then every function in C0(X) which can be approximated on every pair of points in X by functions in B can be approximated uniformly by functions in B.
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