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

?

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+v2+uv2=2u2+2v2forallu,vV. (b) Using part (a), prove that the norm 1 of Exercise 3 on Rn is not induced by any inner product on Rn when n2. Why doesn't the same reasoning work in the case n=1