Let $x = (x_n)_{n \in \mathbb N} \in l^\infty$ and let $T_x : l^1 \rightarrow \mathbb F$ be defined by $T_x (y) = \sum_{n=1}^\infty x_ny_n$. What condition on $x$ is needed so that there exists $y \in l^1$ such that $ \lVert y \rVert_1 = 1$ and $ \lvert T_x (y) \rvert = \lVert T_x \rVert$?
Solution:
$ \lvert T_x (y) \rvert = \lvert \sum_{n=1}^\infty x_ny_n \rvert \le \lVert x \rVert_\infty \cdot \lVert y \rVert_1$ for all $x \in l^\infty, y \in l^1$
For $ \lVert y \rVert_1 = 1$ we have $\lvert \sum_{n=1}^\infty x_ny_n \rvert \le \lVert x \rVert_\infty = sup_{n \ge 1} \lvert x_n \rvert$
I cannot see what condition on $x$ there is for $\lvert T_x (y) \rvert = \lVert T_x \rVert$.