We are supposed to show that the standard sup-norm $\max_{x\in[0,1]}|p(x)|$ and $\max|a_k|$ (largest coefficient) are equivalent over the polynomials of degree n. We must find constants to show this, which may depend on n.
$$c_1(n)\max|a_k|\le \max_{x\in[0,1]}|p(x)|\le c_2(n)\max|a_k|$$
$c_2(n) = n$ follows quite nicely using triangle ineq. and $x^n\le1:x\in[0,1]$
How do I find $c_1(n)$? I don't see $\ge$ inequalities for $\max_{x\in[0,1]}|p(x)|$.
Or would it be easier to swap and put the coefficient norm in the middle?