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Given two orthonormal sets $\{e_k\}_{k=1,2\ldots}$, $\{e'_k\}_{k=1,2\ldots}$ in a Hilbert space $H$, which satisfy $$ \sum_{k=1}^\infty \|e_k-e'_k\|^2 < 1. \tag{*} $$ Prove that if $\{e_k\}_{k=1,2\ldots}$ is complete, $\{e'_k\}_{k=1,2\ldots}$ must be complete. Here by complete, we mean $\overline{span\{e_k\}}=H$.

I tried to prove that it if there is a non-zero $x \perp \overline{span\{e'_k\}}$, i.e. $$\forall k,\quad \langle x,e'_k \rangle = \sum_{j=1}^\infty \langle x,e_j \rangle\langle e_j, e'_k \rangle = 0, $$ then $(*)$ would not hold. But I have not reached any useful results in that direction yet.

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Assuming $\{ e_n \}$ and $\{ e_n'\}$ are orthonormal, define $$ Lx = \sum_{n=1}^{\infty}(x,e_n)e_n' $$ The operator $L$ is linear and satisfies $$ \|Lx\|^2=\sum_{n=1}^{\infty}|(x,e_n)|^2 \le \|x\|^2. $$ Now assume that $\{e_n\}$ is a complete orthonormal basis, and assume that $d=\left(\sum_{n=1}^{\infty}\|e_n-e_n'\|^2\right)^{1/2} < 1$. Then $x=\sum_{n=1}^{\infty}(x,e_n)e_n$ and \begin{align} \|(I-L)x\|^2&=\|\sum_{n=1}^{\infty}(x,e_n)(e_n-e_n')\|^2 \\ &\le \left(\sum_{n=1}^{\infty}|(x,e_n)|\|e_n-e_n'\|\right)^{2} \\ & \le\sum_{n=1}^{\infty}|(x,e_n)|^2\sum_{n=1}^{\infty}\|e_n-e_n'\|^2 \\ & = \|x\|^2\sum_{n=1}^{\infty}\|e_n-e_n'\|^2 = d^2\|x\|^2. \end{align} Because $d < 1$ by assumption, then $\|I-L\| \le d < 1$. It follows that $$ L = I-(I-L) $$ is invertible. Therefore $L$ is surjective, which implies that the closure of the linear space spanned by $\{ e_n'\}_{n=1}^{\infty}$ is $H$. Hence, $\{ e_n'\}_{n=1}^{\infty}$ is a complete orthonormal basis.

Note: I do not know if this is the standard proof for your problem, but I like the proof, and I found on this site. Unfortunately, I cannot find it again. If this is not the standard proof, it should be. :)

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