= Polynomial coefficient sum norm
{title2=$\|p\|_{\mathrm{coef},1}=\sum_{k\geq0}|a_k|$}
For a real or complex <polynomial> $p(t)=\sum_{k\geq0}a_kt^k$, this finite sum defines a <norm>: definiteness follows from uniqueness of <polynomial coefficients>, <absolute homogeneity of a norm> follows termwise, and the <triangle inequality> follows by summing $|a_k+b_k|\leq|a_k|+|b_k|$. On $[0,1]$, its <supremum norm> satisfies $\|p\|_\infty\leq\|p\|_{\mathrm{coef},1}$, since $|t^k|\leq1$. The reverse comparison has no constant independent of the <degree of a polynomial>: for $p_n(t)=(1-t)^n$, the two <norms> are $1$ and $2^n$ respectively, by the <binomial theorem>.
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