Abstract
An approximation process {\left{ {{Γ _n}} \
ight}_{n \in P}} on a Banach subspace X of A [Zemanian A. H. [36]], satisfying either a Jackson type inequality or a Bernstein type inequality of order ρ (n) on X with respect to Y of X, is being related to a class of Banach subspaces {\left{ {{X_λ }} \
ight}_{λ \in J}} of A, on each of which, {\left{ {{X_n}} \
ight}_{n \in P}} defines a sequence of multiplier type operators, satisfying the same inequality with same order. Sufficient conditions for {X_λ } \subset A', λ \in J are given. Results are illustrated by examples.
