I was wondering if the span of a countably infinite list ${f_n}$ of orthonormal vectors in $L^2$ is closed. I just came across the first comment on this post Is the direct sum of two orthogonal subspaces well defined in infinite-dimensional vector spaces?. It says says "The direct sum $M \oplus M^\perp$ is well-defined. It need not be equal to $V$, though. If $V$ is complete and $M$ is closed, then you have $V=M \oplus M^\perp$".
I really don't know how to show that the span of these vectors is closed or open. I'll appreciate any guidence.