Convergence of Sum in a Hilbert Space

I am having some issues understanding what I think may be a pretty fundamental fact about bases in a Hilbert space. If I have a basis $\{f_k\}_{k \geq 1}$ for the Hilbert space $L^2(0, 1)$, is it true that$$\sum_{k \geq 1} a_k f_k$$is $L^2$ convergent to some function $f$ for any selection of scalars $a_k \in \mathbb{R}$? Does this hold in general for any basis $\{x_k\}$ of a Hilbert space $H$ with convergence in the norm? If so, why is this the case?