For all $x = (x_1, \ldots, x_d) \in \mathbb{R}^d$, we set
$$\|x\|_1 := \sum_{i=1}^d |x_i|, \quad \|x\|_\infty := \max\{|x_i|, i = 1, \ldots, d\}.$$
Show that for all $x \in \mathbb{R}^d$, we have
$$\|x\|_1 = \max\left\{x \cdot y, y \in \mathbb{R}^d, \|y\|_\infty \leqslant 1\right\},$$
and
$$\|x\|_\infty = \max\left\{x \cdot y, y \in \mathbb{R}^d, \|y\|_1 \leqslant 1\right\}.$$