The $E$ and $K$ functionals for the pair $(X(A),l_\infty (B))$

We prove some exact formulas for the $E$ and $K$ functionals for pairs of the type $(X(A), l_\infty(B))$ where $X$ has the lattice property. These formulas are extensions of their well-known counterparts in the scalar valued case. In particular we generalize formulas by Pisier [4] and by the present author [2].