The analogs of the Korovkin theorems in Banach function spaces (Q2114878)

In the present paper Korovkin type theorems are studied in the setting of Banach function spaces. The authors introduce, for a given Banach function space \(X\), a special subspace \(X^S\) that they denominate the ``subspace generated by the shift operator'' and prove the density of the \(C_0^{\infty}\) in \(X^S\). Then they prove two theorems which are analogous, for this setting, of the well-known Korovkin theorems. They apply their results to several classes of function spaces, including Lebesgue spaces, grand-Lebesgue spaces, Morrey-type spaces and their weighted versions, weak Lebesgue spaces, Orlicz spaces. In my opinion, the paper is interesting but a clear definition of the subspace \(X^S\) is needed.