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Let $f \in C_0(\mathbb{R})$. Consider the set of translates of $f$ $$ A = \{ f_t : t \in \mathbb{R} \}$$ where $f_t(x)=f(x+t), x\in \mathbb{R}$.

I want to show that $A$ is relatively compact in the weak topology on $C_0(\mathbb{R})$.

I have shown that every sequence in $A$ has a weakly convergent subsequence, using the fact that the dual is the space of Radon measures.

If the closed ball in $C_0(\mathbb{R})$ is metrizable in the weak topology (I'm not sure if it is), then my proof is complete. [The problem is that $C_0(\mathbb{R})$ is not reflexive, and the dual is not separable either]

Is there some result on the metrizability of the unit ball in $C_0(\mathbb{R})$? Or is there some other way to show that $A$ is relatively weak-compact?

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Consider the map $\varphi \colon [-\infty, +\infty] \to C_0(\mathbb{R})$, where the latter is endowed with the weak topology, given by

$$\varphi(t) = \begin{cases} 0 &, t = \pm \infty\\ f_t &, t \in \mathbb{R}.\end{cases}$$

Since for $\lambda \in C_0(\mathbb{R})'$ almost the complete mass of $\lvert\lambda\rvert$ is concentrated on a compact subset of $\mathbb{R}$, $\varphi$ is continuous at $\pm\infty$. It is also continuous at all $t \in \mathbb{R}$ by the uniform continuity of $f$. Thus $\varphi([-\infty,+\infty])$ is weakly compact.

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