Dual of a norm Cone.

Question

Problem

[Example 2.25 Taken from Convex Optimization By Stephen Boyd, Lieven Vandenberghe]

In this example, it proves that the dual of a norm cone is the cone of corresponding dual norm. However, it only talks about $Lp$ norm with $p>1$. I think it also applies for $0<p<1$, but I don't know how to prove it. Thanks for any help.