Let $L^2(\mathbb R)$ is an infinite-dimensional complex vector space. Let $A= \{f_n: n\in \mathbb N\}$ be basis in $L^2(\mathbb R).$
Does there exists $f \in L^2(\mathbb R)$ such that $f$ cannot be written as finite linear combiation of $f_i\in A$? In other other words, there exist $f\in L^2$ and there dose not exist any $m\in \mathbb N$ such that $f=\sum_i^m \alpha_i f_i \ (\alpha \in \mathbb C, f_i \in A)$?