Let $e_1, e_2, \ldots$ be an orthonormal base in the separable Hilbert space $\mathcal{H}.$ Let $\psi_1^n, \psi_2^n, \ldots \in \mathcal{H}, n\in \mathbb{N}$ be vectors such that
$\sup_{i} \| \psi_i^{n}-e_i \|\to 0$ when $n\to \infty.$
My question: Is there exists $N$ such that for any $n>N$ vectors $\psi_1^n, \psi_2^n, \ldots$ are strongly linearly independent (i. e. $\sum_{i=1}^{\infty} a_i \psi_i^n=0, n>N$ only if $a_i=0, i=1,2,\ldots$) and span $\mathcal{H}$ (i. e. $\overline{Lin}\{ \psi_1^n, \psi_2^n, \ldots \}=\mathcal{H}$ for $n>N$).
Comment: This is true for finite dimension but I don't know what happens in infinite dimension.
No, it's not true. Let $$ \psi_i^n = \begin{cases} e_i-\frac{1}{n}\sum_{j=1}^n e_j,\quad &1\le i\le n \\ e_i,\quad & i>n \end{cases} $$ Then $\|\psi_i^n-e_i\|\le 1/n$ for all $i$. On the other hand, $$ \sum_{i=1}^n \psi_i^n =0 $$ and the span of $\{\psi_i^n\}$ is orthogonal to the vector $\sum_{i=1}^n e_i$.