= Dual norm supremum need not be attained
{title2=$\|p\|_*=1\quad\text{without a maximizing unit vector}$}
On the real <space of sequences converging to zero> $c_0$, use the <supremum norm> and define the <linear functional>
$$
p(x)=\sum_{n=1}^{\infty}2^{-n}x_n.
$$
The <geometric series> gives $|p(x)|\le\|x\|_\infty$, so $p$ is a <bounded linear functional> and $\|p\|_*\le1$. The <sequence> with its first $N$ entries equal to one and all later entries zero belongs to $c_0$, has unit <supremum norm>, and gives $p(x)=1-2^{-N}$. Hence $\|p\|_*=1$. But if $\|x\|_\infty\le1$ and $x_n\to0$, not all coordinates can equal one or all equal minus one. Since every weight $2^{-n}$ is positive, $-1<p(x)<1$. The <supremum> is approached but never attained on the <unit ball>.
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