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Given a set of equicontinuous and uniformly bounded functions $\{f_n, f\}$ defined over an open connected $\Omega \in \mathbb{R}^n$. Suppose $(f_n)_n$ weakly converges to some $f$ in distribution sense. That is, assume $$f_n \rightharpoonup f \mathrm{~~in~~} \mathcal{D}'(\Omega)$$ as $n \to \infty$.

Show that $f_n$ converges to $f$ uniformly on compact subsets of $\Omega$.

user31899
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  • Is $\Omega$ open? In this case, you could write it as a countable union of compact sets $(C_i)$ and such that $C_i$ is contained in the interior of $C_{i+1}$ . By the diagonal process, you can extract a subsequence which converges on each $C_i$, hence on compact subset of $\Omega$ – Davide Giraudo Apr 07 '15 at 11:47
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    One way would be to prove that from any subsequence you can extract a further subsequence which has the property. This should be doable with Arzela-Ascoli, which will get you uniform convergence to something, and then you can use the weak convergence to try to ensure that the uniform convergence is actually to $f$. – Ian Apr 08 '15 at 01:29
  • I got it. Thanks! – user31899 Apr 08 '15 at 03:40

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