Does the component of a vector in $L_{2}$ need to be within whatever limits I place upon it?

Question

In $L^{2} (a,b)$ let there be a vector $V(x_{1},x_{2},\dots)$. We have $x_{i}$ defined as:

$$x_{i}\ = \ \langle e_{i}|V \rangle$$

Where $e_{i}$ is the orthonormal basis set. Since $V$ is a part of $L^{2}$ $(a,b)$ does it mean $x_{i} \in (a,b)$? If not, what is the significance of $(a,b)$?