Element in the linear span?

Question

Let $H$ be a Hilbert space and $\{x_i\}_{i\in I}$ a countably infinite set. Denote $S:=\overline{\operatorname{span}\{x_i\}_{i\in I}}$. If $\sum\limits_{i\in I}|{a_i}|^2<\infty$ is then $$\sum\limits_{i\in I}^{}a_ix_i\in S\quad ?$$

Answer 1

No. For instance, consider $x_0\ne 0$, $x_i=2^i x_0$ and $a_i=2^{-i/2}$. Then $s_n=\sum_{k=0}^n a_ix_i=\frac{\sqrt{2^{i+1}}-1}{\sqrt2-1}x_0$, which does not converge to anything.

Similarly, consider $\{e_n\}_{n\in\Bbb N}$ the canonical Hilbert basis of $\ell^2$, $x_i=2^ie_0+e_i$ and $a_i=2^{-i/2}$, which has the same issue with linearly independent vectors.