The Problem
Suppose that $f\in \ell_3^*$. Show that the series $\sum_{n=1}^\infty f(e_n)^3$ converges.
Discussion This problem is on a practice exam I have for a linear analysis course. My first thought on solving this question was to show that the series $(f(e_k))_{k\in \mathbb{N}}$ was an element of $\ell_3$, using the fact that $f$ was a bounded linear functional. In particular, I thought I would be able to construct an element $x_N=\sum_{i=1}^N\alpha_ne_n$ in $\ell_3$ with $\|x_N\|_3\leq 1$ such that $f(x_N)=\sum_{n=1}^N|f(e_n)|^3$ (or at least up to some constant not depending on $N$). We would then have
$$\sum_{n=1}^N|f(e_n)|^3=|f(x_N)|\leq \|f\|\cdot \|x_N\|_3 \leq \|f\|. $$ Taking the limit, the proof would be complete. I thought at least some variant of this argument would work, but I have had no success over the course of a few hours now. If anyone has any suggestions or hints it would be greatly appreciated : )