Comparable norms on the space of polynomials? Are the norms:
$$\|P\|_1=\int^1_0\|P(t) \| dt\mbox{ and }\|P\|_2=\sup_{0\le t\le1} |P(t)|.t$$
comparable on the vector space $X=\mathcal{P}[0,1]$, i.e. all polynomials on $[0,1]$?
Here, i try to use $P_n=\sqrt{n}t^n$ and $Q_n=\sum^n_0 (1-t)^k$ but could no get a result. 
Could you please help.

I am assuming that $||P_1||$ should be $||P||_1$ and similarly for the other norm. Convergence in $||~\cdot~||_2$ is uniform convergence and that is a very strong convergence whereas convergence in $||~\cdot~||_1$ is convergence in $L^1$ and that is a relatively weak convergence. The sequence of standard tent functions all of height 1 do not converge in $||~\cdot~||_2$ but do converge to the zero function in $||~\cdot~||_1$.
