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Linear Algebra : Norms

For any x=(x1,....,xn), let us try to define a norm two ways. Consider

(a) ||X||1=summation |Xi| from i=1 to n

(b) ||X||b=summation |xi-xj| from i,j=1 to n

Does either one of these formulas define a norm? If yes, show that all three axioms of norm hold. If no, demonstrate which axiom fails.

Solution Summary

Summations are investigated as to whether they define norms. The solution is detailed and well presented.