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Numerical linear algebra/norms

Based on the parallelogram law, show that the norms ||.||_1 (1-norm) and ||.||_infinity ( infinity or maximum norm) in R^2 are not induced by any inner product.

Parallelogram Law: ||u+v||^2 + ||u-v||^2 = 2||u||^2 + 2||v||^2.

||x||_1: = sum i = 1 to n of |x_i|

||x||_infinity := max ( 1 =< i =< n)|x_i|

Solution Preview

Proof: (by contradiction
We consider u=(1,0), v=(0,1) in R^2. Suppose ||x||_1 and ||x|_infinity are induced by some inner product, then they satisfy the ...

Solution Summary

This solution is comprised of a detailed explanation to show that the norms ||.||_1 (1-norm) and ||.||_infinity ( infinity or maximum norm) in R^2 are not induced by any inner product.

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