In the course slides for the Stanford Convex Optimization course by Boyd it states that if
$$K=\{(x, t) \,|\, ||x||_1 \le t\}$$
then its dual cone is
$$K^* =\{(x, t) \,|\, ||x||_\infty \le t\}$$
Why is that?
If $x = (-t, 0)$ and $y = (t, 0)$ then $x \in K$ and $y \in K^*$ but $x^ty = -t^2$, which is negative, contrary to the definition of a dual cone.