Define $g :\ell_2 \to \mathbb R$ by $g(x)= \sum_{n=1}^{\infty} \frac{x_n}n$. Is $g$ continuous? Define  $g  :\ell_2 \to \mathbb R$ by
$$g(x)= \sum_{n=1}^{\infty} \frac{x_n}n $$ 
Is $g$ continuous? 
I need to solve this but I could not see how to tackle it? any hints or suggestion?

Hint: $g$ is linear. A linear operator is continuous if and only if it is continuous at $x=0$, if and only if it is bounded.
Try to find a constant $C>0$ such that for all $x$:
$$
|g(x)|\leq C \|x\|
$$
