TITLE: Dual norm of the dual norm is the primal norm
QUESTION [11 upvotes]: How do you prove that the dual of the dual norm is the original norm?
This is what I have so far:
If I have $\|y\|_* $ as the dual norm of $\|y\|$, then I know that
$$
\|y\|_* 
= \qquad 
\max\limits_{x} \ x^Ty
\quad \text{subject to} \quad
\|x\| \le 1
$$
In order to take the dual of this I write the Lagrangian as follows:
$$
L(x,u) = - x^Ty + u\cdot(\|x\| -1)
$$
I rewrote this as
$$
L(x,u)
= - x^Ty - u + u \cdot \sqrt{\sum x_i^2}
$$
Now, taking the dual of this by minimizing the Lagrangian we get
$$
\|y\|_{**}
= \min_{x} L(x,u).
$$
I am not sure how to do this minimization.

REPLY [3 votes]: By Hahn-Banach theorem in the answer above, $$\Vert y \Vert = \max_{x\not=0}\frac{x^T y}{\Vert x \Vert_*} = \Vert y \Vert_{**}$$ Done. I think it's enough if the Hahn-Banach theorem in the answer above is used in a correct way.