Dimension of a subset of $\ell_2$ I am wondering whether the following space is a finite-dimensional subset of $\ell_2$ or an infinite-dimensional one?
$$S(\ell_2)= \{ (x_1,x_2,\ldots) \in \ell_2 \mid \sum_{n=1}^\infty (nx_n)^2 <\infty \}$$
Notice that operator $L(x_1,x_2,...)=(x_1/1,x_2/2,...)$ is a compact one to one operator and thus it can not have an infinite dimensional range. Hence, the dimension of $S(\ell_2)$ comes into question.

Let $e_k = (0,0,0,\ldots,0,0,1,0,0,\ldots)$ where the $1$ is in the $k$th place and all other entries are $0$.  This belongs to your proposed space.
The set $\{e_k : k=1,2,3,\ldots\}$ is linearly independent.
