Prove Equivalent Norms: Norm 1 & Norm 2

[Oster]

Prove that two norms ||.||1 and ||.||2 are equivalent if and only if there exist 2 constants c and k such that c*||x||1 <= ||x||2 <= k*||x||1 for all x in the concerned vector space V.

Attempt-> Equivalence implies a ball in norm 1 admits a ball in norm 2 and vice versa. For normed linear spaces, I know that B(x,r) = x + r*B(0,1).

So, a ball with respect to norm 1, B1(x,r), admits a ball in norm 2 with say radius 's'.

Using the normed linear space property, I can conclude that for a vector 'y' in V, if ||y||2 < s
then ||y||1 < r.

I don't know where I am going =(

 

[micromass]

Hi Oster!

What is your definition of equivalent norms?

 

[Oster]

HI! I got it =D

My definition was that for every open ball with respect to norm 1, there existed an open ball w.r.t norm 2 contained in it and vice versaaaaaaa!

 

[Oster]

Converse was easy pffff.

 

[HallsofIvy]

That is the same as saying that a sequence converges in one norm if and only if it converges in the other.

 

[Gib Z]

In another thread last night you were studying the analogous result for equivalent metrics. This follows from that result, as the norm induces a metric.