Question -
Show that all norms on Rn are equivalent in the sense that, given two norms, say ||·||a and ||·||b, there exist positive constants m and M such that
m||x||a ≤ ||x||b ≤ M||x||a, for all x ∈ Rn.
Hint: The purpose of this problem is to demonstrate that norm equivalence in finite dimensional spaces is really an optimization theoretic property. We will base the proof on Proposition for Euclidean n-space. For an arbitrary norm II - II, consider the functional Φ(x) = ||x|| on the set S = {x: ||x||2 = 1}. Here, ||·||2 is the Euclidean norm. (Show that Φ is continuous in the 2-norm and that S is closed and bounded in the 2-norm, and therefore by Proposition Φ must have a minimize and maximizer on S.)
Use the result above to show that if an operator from Rn to Rm is continuous in one norm, then it is continuous in all norms.