Consider the following linear space \begin{equation}P([a,b],\mathbb{K}) = \{p:[a,b] \rightarrow\mathbb{K}: p \text{ is a polynomial} \} \end{equation}
and for $p \in P([a,b],\mathbb{K})$ the following norm is defined:
\begin{equation} ||p|| = \max\{|a_{k}| : k=0,\dots,\deg(p)\} \end{equation} When \begin{equation}\sum^{\deg(P)}_{k=0} a_{k}x^{k} \end{equation} Also the common sup norm is defined $||p||_{\infty} = \sup_{x \in [a,b]}|p(x)|$.
For $[a,b] =[0,1]$ I need to show that these two norms are not equivalent. I've tried to find a function such that $N||p|| \leq ||p||_{\infty} \leq M||p||$ cannot hold but so far I didn't succeed. I was thinking of a sequence of polynomials such that it converges to $0$ for the regular sup norm and $1/n$ for the $||.||$ norm. Can anyone help me out?
Consider the sequence $\{p_n\}$ in $P([0,1],\mathbb K)$ defined by $$p_n(x)=\sum_{k=0}^nx^k.$$ Then $\|p_n\|=1$ while $\|p_n\|_\infty=n+1.$