Hi! I want to show that sum(norm(x_i-x_j), 1<=i<j<=n) <= n^2 for any vector space over R or C, where x_i are unit vectors.
I haven't got any idea how to show this. Can somebody please help me?
Banach
This is true if you are using a euclidean or Hilbert space norm, but it need not hold for other norms. For example, in the space with the norm , the four vectors are all distance 2 from each other, so the sum of the squares of the six distances is 24 which is greater than 16.
Here's how to prove the result for a real vector space where the norm comes from an inner product .
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The proof for complex spaces is more or less the same, with complex conjugates inserted where appropriate.