From our textbook goes some statement like this:
... let $X$ be the linear subspace of $l_2$ generated by the vectors $\left\{e_1,e_2,e_3,...\right\}$ ...
Which feels strange to me because I thought $e_1=(1,0,0,...)$, $e_2=(0,1,0,...)$, ... are sufficient to generate the entire $l_2$ space, so the $X$ here is actually just $l_2$ itself. Isn't that true?