If $f \in L^2([0,T])$ then it can be written as
$$
f(t) \triangleq \sum_{i \in \mathbb{N}} c_i e_i(t),
$$
for some sequence $\{c_i\}$ of real numbers and some Schauder basis $\{e_i(t)\}$ of $L^2([0,T])$ of smooth functions.   

My question is there a necessary and sufficient condition on the coefficients $\{c_i\}$ of the basis characterizing any function which is in $L^2([0,T])$ and is cadlag?