This problem has already been solved, the solution is at: Prove that if two norms on V have the same unit ball, then the norms are equal. , my question is in regards to the proof. Particularly where did the idea/motivation to use $\frac{v}{p_{1}(v)}$ come from? What was particular about this random vector from the space that allowed it to work for our purposes? Looking at it, it appears as if it is a vector divided by the norm, but why this? Does it give us something to work with?
I understand the "mechanics" of the solution and how it works out, but the intution behind the choice is passing me.
Best Answer
It's the standard way to create a unit vector, i.e. a vector of length $1$.
We are aiming, by contradiction, that the two norms are not equal, meaning there is some vector where they disagree. But for all we know that vector could be far, far away from the origin. There is, a priori, no reason that such a vector should be anywhere close to the unit ball. And if this vector is not close to the unit ball, you can't really exploit the fact that the unit balls are equal to get a contradiction.
You get a similar problem if the vector is really close to the origin.
In order to leverage that the two unit balls are equal, we need to have a vector living right on the edge. This construction ensures that this happens.