\label{QC3}
\|S(\Theta(f))-\Psi(\zeta)\|&=&\|\Psi\circ {\widetilde }{S}\circ R(\Theta(f))-\Psi(\zeta)\|\nonumber\\
&\le& \|\Psi\|_{\rm Lip}\,\|{\widetilde }{S}\circ R(\Theta(f))-\zeta\|.\end{aligned}$$ Since $\Theta(f)\in F$, we have $R(\Theta(f))\in {\widetilde }{F}$. Moreover, by (\[U6\]), the distribution of $\zeta$ is equal to ${\widetilde }{A}(R(\Theta(f)))$. Therefore we get with probability at least 3/4, $$\|{\widetilde }{S}\circ R(\Theta(f))-\zeta\|\le e({\widetilde }{S},{\widetilde }{A},{\widetilde }{F}),$$ and hence, by (\[QC3\]), $$\begin{aligned}
\|S(\Theta(f))-\Psi (\zeta)\|&\le& \|\Psi\|_{\rm Lip}\, e({\widetilde }{S},{\widetilde }{A},{\widetilde }{F})\\
&\le& \|\Psi\|_{\rm Lip}\,(e_n^{{\rm q }}({\widetilde }{S},{\widetilde }{F})+\delta).\end{aligned}$$ $\Psi (\zeta)$ is a random variable with distribution $\Psi(A)(f)$ – the output of the quantum algorithm $\Psi(A)$ from $F$ to $G$ (compare Lemma 2 of [@Hei01] and the definition before it), an algorithm with not more than $2\kappa n$ queries. This implies (\[U3\]).

We need some further notation. For a linear space $X$ we denote by $X^{\#}$ the algebraic dual, that is, the space of all linear (not necessarily continuous) functionals on X, and by $X^*$ the dual space, which is the space of all continuous linear functionals on $X$. Given a subset $F_0$ of a normed space $X$ and $\delta>0$, we denote by $F_0^\delta$ the closed $\delta$-neighbourhood of $F_0$, that is, the set $$F_0^\delta=\cup_{x\in F_0} B(x,\delta),$$ with $B(x,\delta)$ being the closed ball of radius $\delta$ around $x$. The unit ball $B(0,1)$ of $X$ is 