Is $P_n[0,1]$ and $P[0,1]$ are closed in $C[0,1]$

Question

Let $C[0,1]$ be a normed space with supremum norm, then with usual notation, Are the subspaces $P_n[0,1]$ and $P[0,1]$ closed in $C[0,1]$.

I think, by Stone- Wiestrauss theorem, $P[0,1]$ is not closed in $C[0,1]$. As there are limit points of $P[0,1]$ which is not in it. But I think $P_n[0,1]$ is closed, How to prove or disprove. Do Taylor's theorem have any role here?

Answer 1

You are right about $P[0,1]$. For example $\sin$ is not a polynomial since its n-th derivatives are all non-zero, but by Stone Weierstrass it is a limit point of $P[0,1]$. $P_n[0,1]$ is closed because it is finite dimensional and thus complete.