Uniform regularization of the problem of calculating the values of an operator

Let $X$ and $Y$ be linear normed spaces, $W$ a set in $X$, $A$ an operator from $W$ into $Y$, and $\mathfrak M$ the set $\mathfrak G$ of all operators or the set $\mathscr L$ of linear operators from $X$ into $Y$. With $\delta\ge0$ we put
$$ \nu(\delta,\mathfrak M)=\inf_{T\in\mathfrak M}\sup_{x\in W}\sup_{\|\eta-x\|_X\le\delta}\|Ax-T\eta\|_Y. $$
We discuss the connection of $\nu(\delta,\mathfrak M)$ with the Stechkin problem on best approximation of the operator $A$ in $W$ by linear bounded operators. Estimates are obtained for $\nu(\delta,\mathfrak M)$ e.g., we write the inequality, where $H(Y)$ is Jung's constant of the space $Y$, and $\Omega(t)$ is the modulus of continuity of $A$ in $W$.