Let $\|\cdot\|$ be a norm on $\mathbb{R}^{n} .$ The associated dual norm, denoted $\|\cdot\|_{*},$ is defined as $$ \|z\|_{*}=\sup \left\{z^{\top} x \mid\|x\| \leq 1\right\} $$
I'm trying to prove $$\|x\|_{**} = \|x\|$$
here says we can prove it by Hahn-Banach theorem. i.e. $$\|y\|=\max _{x \neq 0} \frac{x^{T} y}{\|y\|_{*}}$$ But I think by definition, this is what we need to prove. So it seems says 'because this is correct, this is correct'. I think it proves nothing.
Are there any proof without using Hahn-Banach theorem?