K}}$, assume that $F$ is the unit ball of a normed space $X$, $\Lambda \subseteq X^*$, and $S$ is a bounded linear operator from $X$ to $G$. Then for all $n\in {{\mathbb N}}_0$ $$\label{VU1}
e^{{\rm q }}_n(S,F)\le 2 e^{{\rm det}}_n(S,F).$$

Let $\delta>0$. It is well-known (see [@TWW88], Theorem 5.2.1 and Corollary 5.2.1) that there is a nonadaptive deterministic algorithm ${\widetilde }{A}$ using at most $n$ informations such that $$\label{RU5}
 \sup_{f\in F} \|S(f)-{\widetilde }{A}(f)\|\le 2 e^{{\rm det}}_n(S,F)+\delta.$$ Such an ${\widetilde }{A}$ has the following form: There are ${{\lambda }}_0,\dots,{{\lambda }}_{n-1}\in \Lambda$ and a mapping ${\varphi }:{{\mathbb K}}^n\to G$ such that $$\label{RU6}
{\widetilde }{A}(f)={\varphi }(f({{\lambda }}_0),\dots, f({{\lambda }}_{n-1}))\quad (f\in F).$$ Without loss of generality we may assume that $\Lambda_0=\{{{\lambda }}_0,\dots,{{\lambda }}_{n-1}\}\subset X^*$ is a linearly independent set (if not, we pass to an independent subset and omit the rest by suitably modifying ${\varphi }$). Let $\theta$ and $\Theta$ be the mappings which result from the application of Lemma \[lem:2\], case (ii). Put $m'=1$, choose $m''\in{{\mathbb N}}$ such that $|\theta({{\mathbb K}})|\le 2^{m''}$ and let $m=m'+m''$. We represent (as done before, in the proof of Lemma \[lem:1\]) $$\theta={\gamma }\circ{\beta }$$ with ${\beta }:{{\mathbb K}}\to {{\mathbb Z}}[0,2^{m''})$ and ${\gamma }:{{\mathbb Z}}[0,2^{m''})\to {{\mathbb K}}$. Furthermore, we 