Parturbations of orthonormal bases Suppose that $(e_n)_{n=1}^{\infty}$ is an orthonormal basis in a Hilbert space $H$, and let $(f_n)$ be an orthonormal sequence in $H$ such that
$$\sum_{n=1}^\infty \|e_n-f_n\|<\infty.$$
How can one show that $(f_n)_{n=1}^{\infty}$ is also a basis in $H$.
 A: Choose $N\in \mathbb{N}$ so that $\sum_{n\geqslant N} \|f_n-e_n\|^2 < 1$. We will show that $(f_k)_{k\geqslant N}$ is a basis for $\{e_k\colon k<N\}^\perp$, which will be enough. Let $f\in \{e_k\colon k<N\}^\perp$ and suppose that $(f,f_{n})=0$ for all $n\geqslant N$. I claim that $f=0$. 
Assume not. Then
$$
\|f\|^{2}=\sum_{n\geqslant N}|(f,e_{n})|^{2}=\sum_{n\geqslant N}|(f,e_{n}-f_{n})|^{2}
\leqslant \sum_{n\geqslant N}\|f\|^{2}\|e_{n}-f_{n}\|^{2}< \|f\|^{2},
$$
which conspicuously yields a contradiction. We have then proved that $(f_k)_{k\geqslant N}$ is a basis for $\{e_k\colon k<N\}^\perp$.
Now, as $(f_k)_{k=1}^\infty$ is an orthogonal sequence, $$H = \overline{{\rm span}}\{e_k\colon k<N\}\oplus  \overline{{\rm span}}\{f_k\colon k\geqslant N\}.$$
By the uniqueness of orthogonal complements, $\overline{{\rm span}}\{e_k\colon k<N\} = \overline{{\rm span}}\{f_k\colon k<N\}$, which proves our assertion.
As a side remark, let me mention a counter-part of this statement for Banach spaces. This is the so-called principle of small perturbations due to Bessaga and Pełczyński:

Suppose that $(e_n)_{n=1}^\infty$ is a basic sequence in a Banach space $X$ having basis constant $M$ and such that the closed linear span of this sequence is complemented by a projection $P$. If $(f_n)_{n=1}^\infty$ is a sequence in $X$ that satisfies
  $$\sum_{n=1}^\infty \|e_n - f_n\| \leqslant \frac{1}{8M\|P\|},$$
  then $(f_n)_{n=1}^\infty$ is a basic sequence equivalent to $(e_n)_{n=1}^\infty$ and the closed linear span of $(f_n)_{n=1}^\infty$ is complemented in $X$.

