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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.

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2 Answers 2

up vote 1 down vote accepted

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.

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

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