Let $(e_j)_1^\infty$ be an orthonormal set in $l^2$
Consider $$s_n =\sum_{j=1}^n t_je_j$$
Show that $s_n$ converges in $l^2 \iff t = (t_j)_{j=1}^\infty \in l^2$
Thoughts so far :
If we consider $T : F \rightarrow l^2$ where $F \subset l^2$ is the space of sequences with Finite Support. $$ T(x) = \sum x_ne_n \:\: \forall x \in F$$ Then it's easy to show that $T$ is an Isometry and since $F$ is dense in $l^2$ , $T$ extends to an Isometry $$T_o : l^2 \rightarrow l^2 $$ $$ T_o(y) = \lim_n T(y_n) \: \: \text{where} \: \: y_n \in F \: \forall n, \: \: \lim_ny_n \rightarrow y$$
I'd like to claim that $s_n$ converges to $T_o(t)$ but I'm sort of stuck on how to proceed.
Is this the right approach?