Summary: This paper discusses pointwise error estimates for the approximation by bounded linear operators of continuous functions defined on compact metric spaces \((X,d)\). The authors introduce a new majorant of the modulus of the continuity which is the smallest among those \(g(\xi)\)’s which have the following properties \(\omega (f,\varepsilon)\leq g(f, \varepsilon)\) and \(g(f, \lambda \varepsilon)\leq (1+ \lambda) g(f, \varepsilon)\) and by this majorant a new quantitative Korovkin type theorem on any compact metric space is proved.