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Unformatted text preview: k x k = < x , x > 1 / 2 . 1 2 (a) Prove that for any x , y ∈ R n , < x , y > = 1 4 ( k x + y k 2 k xy k 2 ) . (b) Prove that k · k p does not derive from an inner product, unless p = 2, that is to say, there does not exist an inner product in R n such that k x k p = < x , x > 1 / 2 ∀ x ∈ R n , unless p = 2. 8. Given a set A ⊂ R n , and x ∈ R n , deﬁne the distance from x to A by d ( x , A ) = inf ± k xy k ² ² ² ² y ∈ A ³ . Prove that for a closed set A , x ∈ A if and only if d ( x , A ) = 0....
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 Fall '09
 Math, Topology, lim, Xn

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