Best approximations of functionals on certain sets

S. B. Stechkin's problem concerning the best approximation of an operator $U$ by bounded linear operators is investigated for the case in which $U$ is a functional. An upper bound is found for the discrepancy of the best approximation and properties of best approximating functionals are investigated. The results are used to study certain functionals related to the problem of finding the best approximation $E_N$ of the differentiation operator in $C(S)$, and the value of $E_N$ is calculated for all cases in which the exact value of the constant in the corresponding Kolmogorov inequality is known.