Let V be an unital Banach algebra, v $\in V$. The smallest subalgebra of V, with contains v and e, is defined as follows:
$\overline{alg(v)}:=\overline{\{\sum\limits_{k=0}^n\lambda_kv^k: \lambda_k\in\mathbb{C}, n\in\mathbb{N}_{0}\}}.$
My question is: Why is it necessary to take the closure of alg(v) to get the completeness of alg(v) / Why alg(v) is not normclosed in V in general?
I know that a closed subspace of a Banach space is itself a Banach space. But in alg(v) are contained finite sums in v with complexe coefficients...I don't see the problem.. Regards