# 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.

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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

$$\frac{1}{n+1} = \|P\|_1 \ge C \|P\|_2 = C$$

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 $||~\cdot~||_1$.