This question may be worded a bit vaguely, but I would like to understand the relevance of the notion of Schauder basis. This notion seems to be of particular interest to Banach space theorists, but is less known outside this field.
My question concerns more specifically the Faber-Schauder basis of $C[0, 1]$. It seems to have no interesting property, apart from being monotone (for example, it is neither shrinking, nor boundedly complete). In all the books I have looked through, this basis is introduced but then nothing is done with it. So, what interesting properties of $C[0, 1]$ can be deduced from the existence of this basis? Are there problems outside the theory of Banach spaces that can be solved by using this basis?