Let $(\mathbb R[X],\|\cdot \|_p)$
I don't really know what $\|P\|_{p}$ stands for a polynomial of $P\in \mathbb R[X]$, and neither do I know what is $\|P\|_\infty $
Is it $$\|P\|_\infty =\sup_{x\in [0,1]}|P(x)| \ \ ?$$ $$\|P\|_2=\sqrt{\int_0^1 |P(x)|^2dx} \ \ ?$$ Furthermore, for the $$\|P\|_p$$, I really have no idea.
On
It could be $L^p$ norm defined as $$ ||P||_p=\sqrt[p]{\int_S|P(x)|^p dp} $$ Here we consider the space $L^p(S)$, with the normal Lebegue measure, where $S\subseteq \mathbb{R}$. If $S=[0,1]$ then this is always a finite value, and the $||P||_2$ and the $||P||_\infty$ norms are exactly what you wrote. If on the other hand $S=\mathbb{R}$, all of the norms are infinite since polynomials diverge, and in fact polynomials are not elements of $L^p(\mathbb{R})$. In that case I believe Surb's answer is the more correct one.
In this situation, I suppose that a polynomial $P(x)=a_0+a_1X+...+a_nX^n$ is identified as the vector $(a_0,a_1,...,a_n,0,0,...)$ in $\mathbb R^{\mathbb N} $ so I could imagine that $$\|P\|_p^p=|a_0|^p+\cdots |a_n|^p$$ and $$\|P\|_\infty =\sup_{i\in\mathbb N}|a_i|.$$