Ah that makes sense, regular definition of basis is not much of use in infinite dimension anyways as far as I recall. 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.
Hey bro, I am not a beggar bro. I am just carrying this termite to the hospital. Can you give some food for the road bro, promise I am not a beggar.