Let V be the vector space of all real-valued bounded sequences. Then for $a,b \in V$ $\langle a,b \rangle :=\sum _{n=1}^{\infty } \frac{a(n) b(n)}{n^2}$ defines a dot product. Find a subspace $U \subset V$ with $U \neq 0, U \neq V, U^\bot=0$.
I couldn't find anything that works, thank you in advance. $U^\bot$ is the orthogonal complement of $U$, in case the notation is confusing.