Consider a separable Hilbert space $H$, and $\{g_n\}$ is an orthonormal basis of $H$. Now there is an orthonormal set $\{f_n\}$ that satisfies $\sum_n\|f_n-g_n\|^2<1$. Show that $\{f_n\}$ is also an orthonormal basis of $H$.
I am not sure how to use the condition $\sum_n\|f_n-g_n\|^2<1$. It implies $\|f_n-g_n\|\rightarrow 0$, which looks like $f_n$ and $g_n$ are almost the same when $n$ is large. But I don't know how to utilize that property.
Thanks in advance!
Suppose that $(f,f_{n})=0$ for all $n$. Then $f=0$ because, if not, $$ \|f\|^{2}=\sum_{n}|(f,g_{n})|^{2}=\sum_{n}|(f,g_{n}-f_{n})|^{2} \le \sum_{n}\|f\|^{2}\|f_{n}-g_{n}\|^{2}< \|f\|^{2}. $$ Therefore the orthonormal set $\{ f_{n}\}$ must be complete because there is no non-zero vector $f$ which is orthogonal to every $f_{n}$.