Let $P$ be the space of all polynomials over $\mathbb{R}$ normed by, $\|P\|= \max \{|a_0|,|a_1|,|a_2|,...,|a_n|\}$ where $p(x)=\sum_{k=0}^{n}a_kx^k$. Is this space complete?
Actually this problem is regarding the open mapping theorem. The map from $p$ to $p$ is not open. But this map isn't contradict the open mapping theorem. I thought its because, $P$ is not a Banach space.
$\sum _{k=1}^{n} \frac {x^{k}} {k!}$ is a Cauchy sequence which does not converge.