Equivalence of Norms in n dimensions

We say that two norms, and , on the same vector space are equivalent if there exist such that for every ,

.

In a finite dimensional normed space, all norms are equivalent.

Proof: Suppose is an n-dimensional space with basis elements and that and are norms on .  We know (from an earlier lemma) there exists a such that for any we choose

(Where .)

Now consider and apply the triangle inequality.

Where .  If we then apply our earlier inequality we’re left with

.

If we repeat this process but with and reversed, we achieve the other inequality.  Since our choice of was arbitrary, and since our and don’t depend on our choice of , we see that and are equivalent.  Since these were arbitrary norms, all norms on are equivalent.