Show that all norms on rn are equivalent in the sense that

Question

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 R^m is continuous in one norm, then it is continuous in all norms.