1. EXTENDING THE NORM Question 1. [3, 5, 6 marks] (a) Let 1 < p < 0. In third year Real Analysis course it was shown that (R", || ||p is a normed space. Using this fact prove that the sequence space (lp, || ||p) is a normed space. That is, for any pair of vectors x := (n)-1 and y := (n)-1 with -1 |xn|P < 00, (i) xp = 0 ⇒ x = 0. (ii) ||Ax|p = |A|||x|p. (iii) x + yp < |x|p+y|p. (Prove each of the norm properties.)

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