Is operator $(Tc)(t) = \sum_{1 \le n} c_n t^n$ from $l^1$ to $C[0,1]$ compact?

Question

I found that it is bounded, but have no idea how to prove it being compact. For $c_n=1/n!$ it has equidistant continuity, for $c_n = (-1)^n/(2n)!$ also, but how to prove it in general way? Please, give me an advice

Answer 1

$T$ is not compact since the image of the bounded sequence $(c(k))_k$ in $\ell^1$ defined by $c(k)_n=\delta_{k,n}$ is the sequence $(t^k)_k$ in $C[0,1],$ which has no uniformly convergent subsequence, its pointwise limit being discontinuous.