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.

Your example works. Assume the norms are equivalent. Then there exists a constant $ C> 0 $ s.t.
$$\|P\|_1 \ge C \|P\|_2 $$
for all $ P$.
Let $ n $ be large enough such that $1/(n+1)< C $. Set $ P (t)=t^n $. Then
$$\frac{1}{n+1} = \|P\|_1 \ge C \|P\|_2 = C $$
Contradiction.
