It is known (and easy) to prove that if $T: H\longrightarrow H $ is compact, where $H$ is a Hilbert space, then for any orthonormal basis $ e_n $ we have $||Te_n||\longrightarrow 0$.

My question is the following: Let $P$ be a orthogonal projection on $H$. Let $ e_n $ be a fixed orthonormal basis such that $$||Pe_n||\longrightarrow 0$$ Does this guarantee that $P$ is compact, i.e. finite rank?