TITLE: Existence of a sequence that converges to a polynomial
QUESTION [5 upvotes]: Let $P\in \mathbb{R}[X]$
Is there a norm $\|\cdot\|$ such that the sequence $(X^n)_{n\in \mathbb{N}}$ which converges to $P$?
Thank you

REPLY [4 votes]: You could construct such a norm as follows. Say $P$ is of degree $\leq d$ (a precaution I only include so as not to have to deal with the zero polynomial in a special case.) Then the family 
$$\mathcal B=(1,X,\dots,X^d,X^{d+1}-P,\dots,X^{n}-P,\dots)$$
is a basis for $V=\Bbb R[X]$. Now define a norm on $V$ by
$$||Q||=\sum_{i=0}^{\infty}\frac{|a_i|}{2^i}$$
where $(a_i)\in\Bbb R^{(\Bbb N)}$ is the (finite) sequence of coordinates of $Q$ with respect to the basis $\mathcal B$. Then by construction, for $n>d$,
$$||X^n-P||=2^{-n}$$
which tends to zero, i.e. $X^n$ tends to $P$.