regular definition of basis is not much of use in infinite dimension anyways as far as I recall.

yeah, that’s exactly why we have an alternative definition for that :D

Wonder if differentiability is required for what you said since polynomials on compact domains (probably required for uniform convergence or sth) would also work for cont functions I think.

Differentiability is not required; what is required is a topology, i.e. a definition of convergence to make sure the infinite series are well-defined.