Let $V$ be a Banach space. A Schauder basis of $V$ is a sequence $(v_n)$ in $V$ such that for each $v$ in $V$ there is a unique sequence $(a_n)$ of complex numbers such that $v=\sum_{n=1}^\infty a_nv_n$.
Now suppose that there is a linearly independent countable set $S\subseteq V$ such that $\text{span}(S)$ is dense in $V$.
My question is if one can necessarily find a Schauder basis of $V$ in $\text{span}(S)$.
Best Answer
Let $V=C[0,1]$ with sup norm and $S=\{1,x,x^{2},...\}$. Then your conditions are satisfied. If there is a sequence of polynomials $p_n$ such that every continuous function is a sum $\sum a_np_n$ (the series converging in sup norm) then every continuous function would be analytic, in particular a $C^{\infty}$ function, which is false.