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The Krein-Smulian theorem is indeed decisive here, thanks! Thanks also to @JochenWengenroth for completing the proof after knowing that the continuous linear forms coincide.
@MichaelGreinecker this is an interesting statement. But then if I have another topology $\tau$ finer than the weak*-topology $\tau_*$, I can deduce that the traces of $\tau$ and $\tau_*$ on the unit ball coincide. But not necessarily that $\tau$ and $\tau_*$ coincide as topologies on $\mathcal{X}*$, right?
@ChristianRemling: you are right, we actually want a vector space topology that makes, for instance, continuous and functional via elements of $\mathcal{X}$ plus possibly one additional functional via an element of $\mathcal{X}'' \backslash \mathcal{X}$$ (as in Nik Weaver approach, see below).
About the last point, we see that the sequence $u \in \ell_1$ should satisfy that $\langle u , e_n \rangle = 0$ for any $n$ and $\langle u, 1 \rangle = 1$, the two requirements being impossible. This seems to prove that it is not possible to extract from $(u_n)$ a subsequence (of sequences) that converges in $\ell_1$. Do you agree?
