hw1additional - Homework 1, Additional Problem. 1 (1) Let 1...

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Unformatted text preview: Homework 1, Additional Problem. 1 (1) Let 1 < p < a real number and let q be defined by 1 = p + 1 . q 1 a. Let f (t) = p tp + 1 - t. Show (by means of calculus), that f (t) 0 for all t 0. q p q a b. Show that ab ap + bq for all a, b > 0. (Hint: Take t = bq-1 in part a.). c. Show that | n ai bi | ( i=1 (b1 , , bn ) Rn . d. Show that n 1 p n i=1 |ai |p ) p ( 1 n i=1 |bi |q ) q for all a = (a1 , , an ), b = 1 n 1 p n 1 p |ai + bi |p i=1 i=1 p |ai |p + i=1 |bi |p . (Hint: Bound on the left |ai + bi | by |ai ||ai + bi |p-1 + |bi ||ai + bi |p-1 and apply part c. to each of the two sums.) 1 e. Show that d(x, y) = ( n |xi - yi |p ) p is a metric on Rn . i=1 1 ...
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This note was uploaded on 02/05/2012 for the course MATH 703 taught by Professor Schep during the Fall '11 term at South Carolina.

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