Vtyjkyuk

Let $(mathbb R[X],|cdot |_p)$

I don't really know what $|P|_p$ stands for a polynomial of $Pin mathbb R[X]$, and neither do I know what is $|P|_infty $

Is it $$|P|_infty =sup_xin [0,1]|P(x)| ?$$
$$|P|_2=sqrt^2dx ?$$
Furthermore, for the $$|P|_p$$, I really have no idea.

It could be $L^p$ norm defined as
$$
||P||_p=sqrt[p]^p dp
$$
Here we consider the space $L^p(S)$, with the normal Lebegue measure, where $Ssubseteq mathbbR$. 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 $S=mathbbR$, all of the norms are infinite since polynomials diverge, and in fact polynomials are not elements of $L^p(mathbbR)$. 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 $$|P|_infty =sup_iinmathbb N|a_i|.$$