$\{e_n\}$ is an orthonormal basis in Hilbert space. $\{f_n\}$ is an orthonormal system in H, such that $\sum ^\infty_1 \| e_n - f_n \| < 1$. Prove that $\{f_n\}$ is a basis.
It looks natural to use projections here, but I don't understand how to do it.
It is enough to show that if$<v,f_n> =0$ for all $n$, then $v=0$. Suppose that we have such a $v \neq 0$.
By the completeness of $\{e_n\}$ we have $$\|v\|^2 = \sum_{n=1}^{\infty} |<v,e_n>|^2 = \sum_{n=1}^{\infty} |<v,e_n-f_n>|^2 \leq \|v\|^2 \sum_{n=1}^{\infty} \|e_n-f_n\|^2 $$
which is a contradiction unless $v=0$ since $\sum ^\infty_1 \| e_n - f_n \| < 1$.
