i am stumped with the following question : Suppose we have an orthonormal sequence $ \left \{ e_{n} , n \in \mathbb{N} \right.\left. \right \} $ in $ L^2[a,b] $ can we add countable many functions such that we obtain an orthonormal basis [ $ L^2[a,b]=\overline{span\left \{ \left \{ e_{n},n \in \mathbb{N}\left. \right \} \cup \left \{ x_{j},j \in \mathbb{N}\left. \right \} \right. \right. \right.} $ ] for $ L^2[a,b] $ ?
Can we generalize for any separable Hilbert space ?
Yes, let $H$ be a separable Hilbert space with dense sequence $f_n$. Suppose you have an orthonormal sequence $e_n$. Let $U$ be the closure of the span of $e_n$ and $U^\perp$ the orthogonal complement of $U$. Let $p$ be the projection onto $U^\perp$.
Do the Gramm-schmidt procedure on $p(f_n)$ to get an orthnormal sequence that lies in $U^\perp$ (thus automatically orthogonal to $e_n$) and has the same (closure of) span as $p(f_n)$. Since the closure of $p(f_n)$ must be $U^\perp$ ($p$ is continuous) you found an orthonormal basis of $U^\perp$ and you started with one of $U$. Together they become a basis of $H$.