Schauder basis for C([0,1])

Question

How can I prove that a system $\{t^k\}_{k=0}^{\infty}$ is not a Schauder basis for C([0,1]) I tried to come up with a function that on the one hand is equal to 1 when t->1, and on the other hand given series $\sum_0^{\infty}{\alpha_k t^k}$ doesn’t converge. And that will contradict a definition of a Schauder basis

Answer 1

If $f \in C[0,1]$ then the following series uniformly converges: $\sum_{k=0}^{\infty}{\alpha_k}$. Therefore: $\sum_{k=1}^{\infty}{k \alpha_k t^{k-1}}$ uniformly converges $\forall t \in (0,1)$. It implies that for any $f \in C[0,1]$ $f \in C^1{[0,1]}$. And it’s clearly not true