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Consider a sequence of finite vectors with length N, i.e. $$\underline{u}_n=\begin{pmatrix}u^1_n\\\vdots\\u^N_n\end{pmatrix},$$ simply for $u^j_n\in \mathbb{R}$. We also have the vector norm $\ell_p$. Do I have some weak convergence results like the $L_p$ spaces? For example if I assume that for this sequence the $\ell_p$-norm is bounded, can I then say that there is limit which $\underline{u}_n$ converges to weakly (like Banach-Alaoglu theorem)?

In any case, I will be grateful if you can introduce a reference with related discussion.

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All norms are equivalent in finite dimensional space, so the balls are compact and you can extract a subsequence of $u_n$ which converges if $u_n$ is bounded

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