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Let $\mathcal{P}$ be the infinite dimensional vector space of polynomials defined on $[0,1]$. Show $||p||_1:= \sup\{|p(x)|:x\in[0,1]\}$ and $||p||_2:= \int_0^1 p(x)dx$ are not equivalent norms on $\mathcal{P}$.

I proved both $(\mathcal{P},||\cdot||_1)$ and $(\mathcal{P},||\cdot||_2)$ are not complete. Can I conclude directly from this fact that the norms are not equivalent? If not, what more do I need to show?

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    Surely you mean $\|p_2\|:=\int_0^1|p(x)|dx$. Anyway, you can show that $p_n(x)=x^n$ is Cauchy in $\|\,\|_2$ but not in $\|\,\|_1$. Can you see how this helps? – 2017-02-19
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    Yes it is explicitly stated like that in the exercise. Usually $||p||_2= (\int_0^1 |p(x)|^2dx)^\frac{1}{2}$ right? I will try $x^n$. – 2017-02-19
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    There are different norms, you can use $\|p\|_2=\left(\int_0^1|p(x)|^q\right)^{1/q}$ for any $1≤q<\infty$ if you want to. These are all inequivalent. – 2017-02-19
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    thank you! With a sequence that is Cauchy in one norm but not in another you have an example wherefore the definition of equivalent norms does not hold. – 2017-02-19

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