There are ways to select two indices . So
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.