to $(E_\Psi^{q'})_t(\rn)$.

Let $t\in(0,\infty)$. Using Proposition \[th2\], we obtain $$(E_\Phi^q)_t(\rn)=\ell^q(\{L^\Phi(Q_{tk})\}_{k\in\mathbb{Z}^n})$$ and, for any $f\in(E_\Phi^q)_t(\rn)$, $$\|f\|_{(E_\Phi^q)_t(\rn)}\sim\frac{t^{\frac{n}{q}}}{[\Phi^{-1}(\frac{1}{\varepsilon_nt^n})]^{-1}}
\|f\|_{\ell^q(\{L^\Phi(Q_{tk})\}_{k\in\mathbb{Z}^n})}.$$ Then it is easy to prove that $((E_\Phi^q)_t(\rn))^*=(\ell^q(\{L^\Phi(Q_{tk})\}_{k\in\mathbb{Z}^n}))^*$ and, for any $f\in((E_\Phi^q)_t(\rn))^*$, $$\label{11}
\frac{t^{\frac{n}{q}}}{[\Phi^{-1}(\frac{1}{\varepsilon_nt^n})]^{-1}}
\|f\|_{((E_\Phi^q)_t(\rn))^*}\sim\|f\|_{(\ell^q(\{L^\Phi(Q_{tk})\}_{k\in\mathbb{Z}^n}))^*},$$ where the equivalent positive constants are independent of $t$ and $f$. Applying [@mmz p.110, Theorem 7] or [@af Theorem 8.19], we know that $(L^\Phi(Q_{tk}))^*$ is isomorphic and homeomorphic to $L^\Psi(Q_{tk})$, which, together with Lemma \[dou\], implies that $(\ell^q(\{L^\Phi(Q_{tk})\}_{k\in\mathbb{Z}^n}))^*
=\ell^{q'}(\{L^\Psi(Q_{tk})\}_{k\in\mathbb{Z}^n})$ and, for any $f\in(\ell^q(\{L^\Phi(Q_{tk})\}_{k\in\mathbb{Z}^n}))^*$, $$\label{22}
\|f\|_{(\ell^q(\{L^\Phi(Q_{tk})\}_{k\in\mathbb{Z}^n}))^*}
\sim\|f\|_{\ell^{q'}(\{L^\Psi(Q_{tk})\}_{k\in\mathbb{Z}^n})},$$ where the equivalent positive constants are independent of $f$ and $t$. Using Proposition \[th2\] again, we find that, for any $f\in \ell^{q'}(\{L^\Psi(Q_{tk})\}_{k\in\mathbb{Z}^n})$, $$\|f\|_{\ell^{q'}(\{L^\Psi(Q_{tk})\}_{k\in\mathbb{Z}^n})}\sim\frac{[\Psi^{-1}
(\frac{1}{\varepsilon_nt^n})]^{-1}}{t^{\frac{n}{q'}}}
\|f\|_{(E_\Psi^{q'})_t(\rn)},$$ which, combined with (\[11\]) and (\[22\]), implies that $((E_\Phi^q)_t(\rn))^*=(E_\Psi^{q'})_t(\rn)$ and $$\frac{t^n}{[\Phi^{-1}(\frac{1}{\varepsilon_nt^n})]^{-1}[\Psi^{-1}
(\frac{1}{\varepsilon_nt^n})]^{-1}}\|f\|_{((E_\Phi^q)_t)(\rn)^*}
\sim\|f\|_{(E_\Psi^{q'})_t(\rn)},$$ where all equivalent positive constants are independent of $f$ and $t$. By Lemma \[un\], we have $$\frac{t^n}{[\Phi^{-1}(\frac{1}{\varepsilon_nt^n})]^{-1}[\Psi^{-1}
(\frac{1}{\varepsilon_nt^n})]^{-1}}
\sim 1$$ with the equivalent positive constants independent of $f$, which further implies the desired conclusion and hence completes the proof of Theorem \[dual\].

Next, we recall the notion of ball quasi-Banach function spaces defined in [@ykds Definition 2.1]. In what follows, the *symbol* $\mathbb{M}(\rn)$ denotes the set of all measurable functions on $\rn$.

\[ball\] A quasi-Banach space $X\subset\mathbb{M}(\rn)$ is called a *ball quasi-Banach function space* on 