Criterion for a relatively shift weakly compact sequence of measures to be actually relatively weakly compact

I'm trying to understand a result given in Theorem 2.3.7 (see below) of Linde's Probability in Banach spaces: Stable and Infinite Divisible Distributions.

Let $$(\mu_n)_{n\in\mathbb N}$$ be a sequence of Radon probability measures on the Borel $$\sigma$$-algebra $$\mathcal B(E)$$ of a Banach space $$E$$ such that there is a sequence $$(x_n)_{n\in\mathbb N}\subseteq E$$ such that $$(\mu_n\ast\delta_{x_n})_{n\in\mathbb N}$$ is relatively compact (denoted by "w.r.c." below) with respect to the topology of weak convergence of measures.

If we denote the topology of compact convergence on $$E'$$ by $$\tau_c$$, we can show that the family $$\{\hat\mu:\mu\in\mathcal F\}$$ of characteristic functions of a bounded (in total variation norm) and tight family $$\mathcal F$$ of finite signed measures on $$\mathcal B(E)$$ is uniformly $$\tau_c$$-continuous.

Assume now that there is a $$\delta>0$$ such that $$C_\delta:=\left\{\left.\hat\mu_n\right|_{V_\delta}:n\in\mathbb N\right\}$$ is $$\tau_c$$-equicontinuous at $$0$$ (which is actually equivalent to being uniformly $$\tau_c$$-equicontinuous, which in turn is equivalent to being relatively compact with respect to the uniform topology on $$C(V_\delta)$$), where $$V_\delta:=\{\varphi\in E':\left\|\varphi\right\|_{E'}\le\delta\}$$.

In conclusion, $$C_\delta$$ and $$\left\{\left.\widehat{\mu_n\ast\delta_{x_n}}\right|_{V_\delta}:n\in\mathbb N\right\}$$ are both $$\tau_c$$-equicontinuous at $$0$$.

This should yield the existence of $$\rho>0$$ and a compact $$K\subseteq E$$ such that $$\left|1-\hat\mu_n(\varphi)\right\|<\frac\varepsilon2\tag1$$ and $$\left|1-e^{{\rm i}\langle x_n,\:\varphi\rangle}\hat\mu_n(\varphi)\right\|<\frac\varepsilon2\tag2$$ for all $$\varphi\in V_\delta$$ with $$p_K(\varphi):=\sup_{x\in K}|\langle x,\varphi\rangle|<\rho$$ and $$n\in\mathbb N$$.

However, as you can see in the proof of Linde below, it seems like he is assuming $$\rho=1$$. That doesn't make sense to me. Why should $$(1)$$ and $$(2)$$ be smaller than $$\varepsilon/2$$ for an arbitrary $$\varepsilon$$ as long as $$p_K(\varphi)\le1$$?

If $$K$$ is a compact subset of $$E$$, then so is $$RK = \{Rx \, \mid \, x \in K\}$$, no matter the choice of $$R > 0$$. Further, given $$a \in E'$$, $$\sup \left\{ |\langle a,x \rangle| \, \mid \, x \in RK \right\} = R \sup \left\{ |\langle a,x' \rangle| \, \mid \, x' \in K \right\}$$. Hence, by dilation, we can replace $$\rho$$ by $$1$$.
• I thought on this problem in a more general setting of continuous functions and completely missed that we are considering linear continuous functions here. So, I simply can replace $K$ by $K/\rho$ ... Jan 13 '21 at 7:56