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Let $\varepsilon \in [0,1)$ and $n\in\mathbb N$. Define $G : \mathbb N\times [0,1) \to\mathbb R^\mathbb R$ s.t $G(n,\varepsilon)\in \mathcal C^n(\mathbb R), n\in\mathbb N$ and \[ G(n,\varepsilon) \xrightarrow[\varepsilon\to 1-]{} 0 \quad \text{in }\mathcal C^n\text{ norm}\qquad (n\in\mathbb N) \] Does $\sum_{n\in\mathbb N} G(n,\varepsilon)\xrightarrow [\varepsilon \to 1-]{} \mathcal F$ uniformly only if $\mathcal F\in\mathcal C^\infty$?
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Thu, 14 Mar 2019 10:03 GMT