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Exercise 4 (Parallelogram law and norms not induced by inner products). (a) Prove that the norm in an inner product space V satisfies the parallelogram
Exercise 4 (Parallelogram law and norms not induced by inner products).\ (a) Prove that the norm in an inner product space
V
satisfies the parallelogram\ law\
||u+v||^(2)+||u-v||^(2)=2||u||^(2)+2||v||^(2), for all u,vinV.
\ (b) Using part (a), prove that the norm
||*||_(1)
of Exercise 3 on
R^(n)
is not induced\ by any inner product on
R^(n)
when
n>=2
. Why doesn't the same reasoning\ work in the case
n=1
?
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