\[pro:5\] Let $X$ and $Y$ be normed linear spaces such that $X$ is a linear subspace of $Y$ and the embedding $J:X\to Y$ is continuous. Assume that $K={{\mathbb K}}$, $F=B_X$, $\Lambda$ is a linearly independent subset of $Y^*$, $T:Y\to G$ is a bounded linear operator, and $S=TJ$. Then for all $n_1, n_2\in {{\mathbb N}}_0$, $$\label{CJ4}
e^{{\rm q }}_{\nu_1 n_1+2\nu_2 n_2}(S,F)
\le 
8 e^{{\rm det}}_{n_1}(J,F)\,e^{{\rm q }}_{n_2}(T,B_Y),$$ where $\nu_1,\nu_2$ are any numbers satisfying (\[KB1\]). If, furthermore, $J$ is a compact operator and $\Lambda$ is linearly independent over $X$, then $$\label{RT3}
e^{{\rm q }}_{\nu_1 n_1+2\nu_2 n_2}(S,F)
\le 
8e^{{\rm det}}_{n_1}(J,F)\sup_{\mathcal{E}\in\mathscr{C}(B_Y)} e^{{\rm q }}_{n_2}(T, \mathcal{E}).$$

Weighted mean computation and integration
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Let $L_1^N$, respectively, $L_\infty^N$, be the space of all functions $f:{{\mathbb Z}}[0,N)\to {{\mathbb K}}$, equipped with the norm $$\|f\|_{L_1^N}=\frac{1}{N}\sum_{i=0}^{N-1}|f(i)|,$$ respectively, $$\|f\|_{L_\infty^N}=\max_{0\le i< N}|f(i)|.$$ We use the notation $L_1^N({{\mathbb K}})$ and $L_\infty^N({{\mathbb K}})$ if the underlying field has to be emphasized. Let $g\in L_1^N$. Define the weighted mean operator $S_{N,g}: L_\infty^N\to {{\mathbb K}}$ by $$S_{N,g}f=\frac{1}{N}\sum_{i=0}^{N-1}g(i)f(i) \quad (f\in L_\infty^N).$$ We write $S_{N}$ for $S_{N,g}$ with $g\equiv 1$. We consider the weighted summation problem $\mathcal{P}=(B_{L_\infty^N},{{\mathbb K}},S_{N,g},{{\mathbb K}},\Lambda)$ with $\Lambda=\{\delta_i\,:\,0\le i<N \}$ and $\delta_i(f)=f(i)$. Throughout this paper we often use the same symbol $c,c_1,\dots$ for possibly different 