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I have a countable infinite normed, equiangular sequence $a_n \in \ell^2$, i.e $\langle a_n, a_m \rangle=\theta$ for $n\not=m$ and $\langle a_m, a_m \rangle =1$ for some $\theta <1$. It's clear that the $a_n$ does not converge. Is it still possible that their duals $\phi_{a_n}=\langle a_n,\cdot \rangle$ converge pointwise, i.e $\phi_{a_n}(x) \rightarrow \phi (x) \quad \forall x\in \ell^2$?

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