Why do polynomial norms only consider the coefficients? Rather than the basis as well?
It seems a bit illogical to say that $\|p\|_1=\sum_{k=0}^n |a_k|$, but e.g. $\|f\|_1=\sum_{j=0}^k \| f^{(j)}\|_{\infty}$ (assuming $f\in C^k$).
Maybe this is just some intuition regarding "what's the important property that we want to measure in the construct". So for polynomials one's often interested in the coefficients (since the basis is known).