Let $P=\sum_{k=0}^n a_k z^k$ be a polynomial of degree $n$. How would you prove that the max of coefficients norm of $P$, i.e. ${\Vert P\Vert}_1=\max_{0\le k\le n}|a_k|$ is smaller than the norm, ${\Vert P\Vert}_2= \sup |P(z)|$ for $z$ belonging to the unit ball?
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