Approximation of the continuous functions on $$l_{p}$$ spaces with p an even natural number

Abstract

Let p be an even natural number, $$K\subset l_{p}$$ a compact nonempty set and $$L_{n}:C\left( K\right) \rightarrow C\left( K\right) $$ a sequence of positive linear operators. We prove that, under suitable assumptions, for every $$f\in C\left( K\right) $$, $$\lim \nolimits _{n\rightarrow \infty }L_{n}\left( f\right) =f$$ uniformly on K. As applications we give infinite variants of the Bernstein and Kantorovich theorems.