= Norm attainment on the vanishing sequence space
{title2=$f_a\text{ attains its norm}\iff a\in c_{00}$}
The <continuous linear functional> $f_a(x)=\sum a_nx_n$ on the real <space of sequences converging to zero> corresponds to $a\in\ell_1$ and has <operator norm> $\sum|a_n|$. Equality $f_a(x)=\|f_a\|$ on the closed unit ball forces $x_n=\operatorname{sgn}(a_n)$ on the support of $a$, since every term $|a_n|-a_nx_n$ is nonnegative. Such an $x$ can tend to zero only if that support is finite. Conversely, the finite sign sequence attains equality for every <finitely supported sequence> $a$.
Back to article page
