Could we say that the set $\{1,x,x^2,x^3,x^4,\cdots\}$ is a basis of the vector space of polynomials ?
Note that, for example the set $\{e_1,e_2,e_3,\cdots\}$ such that $e_n=(\delta_{n,k})_{k \in \mathbb{N^*}} $ is not a basis for the vector space $\mathbb{R}^\infty$, even though the set $\{e_1,e_2,e_3,\cdots,e_n\}$ is a basis of $\mathbb{R}^n$ for any $n \in \mathbb{N}^*$.