In Brezis's Functional Analysis, theorem $4.25$, he gives a $L^p$ version of Arzela-Ascoli theorem as follows
Let $\mathcal{F}$ be a bounded set in $L^p(\mathbb{R}^n)$ with $1\leq p<\infty.$ Assume that $$\lim_{|h|\to 0}\left\|\tau_{h}f-f\right\|_{p}=0\qquad\text{uniformly in } f\in\mathcal{F}\tag{1}$$ ($\tau_{h}f$ is the left translation of $f$ by $h$) then the closure of $\mathcal{F}\big{|}_{\Omega}$ in $L^{p}(\Omega)$ is compact for any measurable set $\Omega\subset\mathbb{R}^N$ with finite measure (here $\mathcal{F}\big{|}_{\Omega}$ denotes the restrictions to $\Omega$ of the functions in $\mathcal{F}.$)
now I want to recover it for $1<p<\infty$ via the following method.
Denote $E=L^{q}(\mathbb{R}^N)$ with $\frac{1}{p}+\frac{1}{q}$ and $B_{E}$ the closed unit ball in $E,$ since $E$ is reflexive, $B_{E}$ is compact in the weak topology $\sigma(E,E^{\ast}),$ moreover, since $E^{\ast}=L^{p}(\mathbb{R}^N)$ is separable, we derive that $B_E$ is metrizable in $\sigma(E,E^{\ast})$ and the metric on $B_E$ can be define by $$d(g_1,g_2)=\sum_{n=1}^{\infty}2^{-n}\left|\int_{\mathbb{R}^N}f_n(g_1-g_2)\right|$$ where $\left\{f_n\right\}\subset L^{p}(\mathbb{R}^N)$ is any dense subset of the closed unit ball in $L^p(\mathbb{R}^N)$ (ref: Haim Brezis, Functional Analysis, theorem $3.29$), to summarize: $B_E$ is a compact metric space with the subspace topology of the weak topology of $E.$
Since every $f\in L^{p}(\mathbb{R}^N)$ can be regarded as a continuous function on $B_E,$ i.e. $$f:B_E\to\mathbb{R},\left\langle f,g\right\rangle=\int_{\mathbb{R}^N}fg,\forall g\in B_E$$ and the uniform norm of $f$ will be $$\left\|f\right\|_{u}=\sup_{g\in B_E}\left|\left\langle f,g\right\rangle\right|=\sup_{g\in L^{q}(\mathbb{R}^N),\left\|g\right\|_{q}\leq 1}\left|\int_{\mathbb{R}^N}fg\right|=\left\|f\right\|_{p}$$ so apply the classical Arzela-Ascoli theorem we derive that any $\mathcal{F}\subset L^p(\mathbb{R}^N)$ has compact closure with respect to the uniform norm on $C^0(B_E)$ (and thus with respect to the $L^p$ norm) iff $\mathcal{F}$ is bounded and equi-continuous, i.e. $$\forall\varepsilon>0,\exists \delta>0,\forall g_1,g_2\in B_E, d(g_1,g_2)<\delta,\text{we have } |\left\langle f,g_1-g_2\right\rangle|<\varepsilon,\forall f\in \mathcal{F}\tag{2}$$ so how can I recover $(1)$ from $(2)$ so as to get the $L^p$ version of Arzela-Ascoli theorem? Any help is appreciated!