I'm asked to give an example of an $x\in \mathcal{l}^2$ s.t.
$$\sum_{j=1}^{\infty}|<e_j,x>|^2<||x||^2$$
Where $(e_j)$ is some orthonormal sequence.
However, I think this question is more about finding a sequence $(e_j)$ s.t. there exist $x$ for which strict inequality hold then about finding an $x$ for which strict inequality holds. Because finding such $x$ is not always possible, most notably this is (I believe) impossible when the $(e_j)$ form a total basis for $\mathcal{l}^2$. So asking to find such $x$ with no regard to what sequence $(e_j)$ we are working with seems strange?
More concretely, if we take $(e_j)=\delta_j$ then we have:
$$\sum_{j=1}^{\infty}|<e_j,x>|^2=\sum_{j=1}^{\infty}|\sum_{i=1}^{\infty}e_{j_i}\bar{x_i}|^2=\sum_{j=1}^{\infty}|x_j|^2=||x||^2$$
So that we will always have equality. However if we take a sequence like $(e_j)=\delta_{2j}$ then we will have inequality for every $x$ that has a non-zero odd term.
So I'm asking if I'm correct that it makes little sense to ask for an $x$ for which inequality hold with no regard for the sequence $(e_j)$?
If $e_j$ is an orthonormal basis, then you will always have equality, by definition. The only way to achieve inequality is if $e_j$ is orthonormal but not a basis, for example as you did by just taking even terms.