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.(adsbygoogle = window.adsbygoogle || []).push({});

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 im going =(

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# Homework Help: Equivalent Norms =(

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