Sequence in Hilbert space whose inner product with all basis vectors goes to 0

Question

I'm trying to solve a problem involving a Hilbert space $H$ and corresponding Hilbert basis $\mathcal B$ where $\{x_n\} \subset H$ is a sequence satisfying $\lim_{n \to \infty} \langle x_n, b \rangle = 0$ for all $b \in \mathcal B$. But I can't think of any examples of such a space and basis and sequence except the trivial case where $x_n \to 0$. Can someone please give me a simple example? Say on $\mathbb R^d$, or on one of the other standard Hilbert space examples studied in introductory functional analysis.

Answer 1

Take any Hilbert space with a countable orthonormal basis $(e_n)$ and take $x_n=e_n$ for all $n$.

Answer 2

In every infinite-dimensional Hilbert space, all of the orthonormal sequences are convergent to zero weakly, i.e. if $(x_n)$ is an orthonormal sequence, we have that $$(x_n\mid a) \to 0\quad \forall a \in H$$ This is a consequence of Bessel's inequality: Let $(e_n)$be an orthonormal sequence in a Hilbert space $H$. Then the following inequality holds for all $x$: $$\sum_{n=0}^{+\infty}|(x\mid e_n)|^2\leqslant \lVert x \rVert ^2$$ Which forces the general term to go to zero, i.e. $|(x\mid e_n)|^2 \to 0$.