Check if following funtional is continuous
$ R[X]\ni p \to p(0)p \in R[X] $
towards norm:
$||p|| = \sum_{n=1} 2^{-n}|p(\frac{1}{n})|$
I've started:
$||p|| = sup_{\sum_{n=1} 2^{-n}|p(\frac{1}{n})|\le1}\{\sum_{n=1} 2^{-n}|p(0)p(\frac{1}{n})|\} \le p(0)$
But I'm not able to give a counterexample, which shows that this functional is discontinuous indeed.