Almost orthonormal projection and orthonormal projection in Hilbert space Let $(e_i)_i$ be a family of vectors in a Hilbert space being almost orthonormal but not quite, i.e.
$$\langle e_i, e_j \rangle  \approx  \delta_{i,j} + \alpha e^{-\vert i-j \vert} $$
and $\alpha$ is a small positive number. 
It is tempting then to say that 
$$Q:=\sum_{i=1}^{\infty} \langle \bullet, e_i \rangle e_i$$ 
is almost the orthogonal projection onto the closed span of the $e_i$ which we denote by $P.$
However, this is not quite right, as the basis is not fully orthonormal, so is there an estimate on the operator norm
$$\Vert Q-P \Vert ?$$
It seems that an error $$\alpha \lesssim \Vert Q-P \Vert  $$ is unavoidable (just by testing both against one of the $e_i$.
But is there also a bound from above?
 A: Indeed, there is also a bound $\|P-Q\| \lesssim \alpha$.
There are certainly many other more involved and elegant proofs, but here is a very basic one. As noticed in the comments, we can assume that the $e_i$ span a dense subspace, in which case we have to prove that $\|Q-Id\|\lesssim \alpha$. In other words that for every $x = \sum_i \lambda_i e_i$ in the linear span, we have $\|Qx-x\| \lesssim \alpha \|x\|$.
The right-hand side is equal to the square root of
$ \sum_{i,j} \lambda_i \overline{\lambda_j} \langle e_i,e_j\rangle$,
so by the assumption and Cauchy-Schwarz we get
$$ |\|x\|^2 - \sum_i |\lambda_i|^2 |\leq \alpha \sum_{k \in \mathbb Z} e^{-|k|} \sum_i |\lambda_i \lambda_{i+k}|  \leq \alpha \sum_{k \in \mathbb Z} e^{-|k|} \sum_i |\lambda_i|^2.$$
So for $\alpha$ small enough we have
$$ \|x\| \lesssim (\sum_i |\lambda_i|^2)^{\frac 1 2} \lesssim \|x\|.$$
Similarly, $Qx-x= \sum_{i,j} \lambda_i (\langle e_i,e_j\rangle e_j - \delta_{i,j}) e_j$, so by the preceding inequality for $Qx-x$,
$$\|Qx-x\| \simeq  (\sum_{j} |\sum_i \lambda_i(\langle e_i,e_j\rangle  - \delta_{i,j})|^2)^{\frac 1 2} \lesssim \alpha (\sum_{j,s,t} |\lambda_s \lambda_{t}| e^{-|s-j|-|t-j|})^{\frac 1 2}.$$
Bound $\sum_j e^{-|s-j|-|t-j|} \lesssim e^{-|s-t|/2}$. You obtain
$$ \|Q_x-x\|\lesssim \alpha (\sum_{s,t} |\lambda_s\lambda_t| e^{-|s-t|/2})^{\frac 1 2} \lesssim \alpha (\sum_i |\lambda_i|^2)^{\frac 1 2}.$$
This is $\lesssim \alpha \|x\|$ by the first computation.
