I have the following questions about dual norms :

How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far:

If I have $||y||_* $ as the norm dual of $ || y ||$ then I know that $\\$

$||y||_* $ = $max_x \ x^Ty $ subject to $ ||x|| \leq 1 $

In order to take the dual of this I first write the Lagrangian as follows:

$ L(x,u) = - x^Ty + u*(||x|| -1) $

I rewrote this as:

$ L(x,u) = - x^Ty + u*\sqrt{(\sum x_i^2)} \ - u $

Now, taking the dual of this by minimizing the Lagrangian we get the following :

$||y||_{**} = min_x L(x,u)$

I am not sure how to do this minimization. I would also like to confirm that all the former steps are correct. I understand that this is probably fairly simple - but I'm fairly new to this and any help would be very appreciated.