I read an answer in the post that $\{1,x,x^2,\cdots\}$ forms a basis for $\mathbb{C}[x]$. I want to extend it a bit more general:
Is it true that if $p_0,p_1,\dots$ are any polynomials from $\mathbb{C}[x]$, then $\{p_0,p_1,\dots\}$ forms a basis for $\mathbb{C}[x]$ as well, or does it also require that $\deg p_n=n$?
No, arbitrary countable subset of polynomials does not necessarily form a basis. If each degree of $p_n = n$, it is true that it's a basis.