Define $f : L^2 \rightarrow \mathbb{R}$ by $f(x) = \sum_{n=1}^{\infty} \frac{x_n}{n}$. Is $f$ continuous? Problem I came across studying for qualify exams. I think I have it, if $x^k$ is a sequences of sequences in $L^2$ and it converges to $x$ in $L^2$, then $d(x^k_n, x_n) \rightarrow 0$ as $k \rightarrow \infty$ for all $n$. Then consider $d(f(x^k), f(x)) = \sum_{n=1}^{\infty} \frac{x^k_n - x_n}{n}$. But since $d(x^k_n, x_n) \rightarrow 0$ as $k \rightarrow \infty$, $d(f(x^k), f(x)) \rightarrow 0$ as $k \rightarrow \infty$. Is this right? Also, I was curious if it's even true that $f(x)$ is finite for all $x \in L^2$. I'm suspicious that it's not necessarily true but I couldn't figure out a proof or find a counterexample. 

Hint:
$$|f(x) - f(y)| \le \frac{\pi}{\sqrt 6} \left( \sum_{k=1}^\infty (x_k - y_k)^2 \right)^{1/2}.$$
