Let $\mathbb{R}^{\mathbb{N}}$ be the space of all real sequences. Let $V$ be the subspace of all bounded real sequences equipped with
$$\langle f,g\rangle=\sum_{n=1}^\infty\frac{f(n)g(n)}{n^2}$$
Find a subspace $U$ of $V$ with $U \neq 0$, $U \neq V$ and $U^ ⊥ = 0$.
So that would mean that the only $g(n)$ which satisfies
$$\sum_{n=1}^\infty\frac{f(n)g(n)}{n^2}=0 $$
for all $f \in U$ would be $g=0$.
So far I have tried testing all sequences that do not converge to $0$ and examining $$\sum_{n=1}^\infty\frac{f(n)}{n^2}$$
Am I on the right track?
I have also found out that there can be no $n \in \mathbb{N}$ such that $f(n)=0$, because you could define $g(n)=1$ for that $n$ and $0$ otherwise.
However my attempts have failed, as it seems to me that for every bounded sequence you can scale the summands by $g(n)$ so that the sum diminishes.
Any hints are welcome.