Is the space of polynomial isometric to $\mathbb R^n$?

Question

Consider the space of polynomials of degree $n-1$ defined on $[0,1]$ with $\| p(x)\| = \max_{x\in[0,1]}\vert p(x) \vert$. Is it isometric to $\mathbb R^n$? Is it possible to find a bijective isometry?

Answer 1

I think you meant polynomial of degree at most $n-1.$ And, I also feel, you meant absolute maximum, to be your norm.

Can of find extreme points of the unit ball of $\mathbb{R}^n$ and the space you mentioned. Note that number of extreme points is "something" preserved under isometric isomorphism.