Uniqueness of dilation invariant norms

Let $\delta_a$ be a nontrivial dilation. We show that every complete norm $\Vert\cdot\Vert$ on $L^1(\mathbb{R} ^N)$ that makes $\delta_a$ from $(L^1(\mathbb{R} ^N),\Vert\cdot\Vert)$ into itself continuous is equivalent to $\Vert\cdot\Vert _1$. $\delta_a$ also determines the norm of both $C_0(\mathbb{R} ^N)$ and $L^p(\mathbb{R} ^N)$ with $1<p<\infty$ in a weaker sense. Furthermore, we show that even all the dilations do not determine the norm on $L^\infty(\mathbb{R} ^N)$.