Let $E=\mathbb{R}[X]$
We define $N:\, P \to \sum_{n=0}^{\infty} { |P^{(n)}(n)|}$ ($P^{(n)}$ being the $n$-th derivative) , it is not hard to prove that $N$ is a norm on $E$.
Help me to study the continuity of the linear form $f_m\colon\, P \to P_{[m]}$ ($P_{[m]}$ being the coefficient of $x^m$ in $P$) for some positive integer $m$.