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# hw3 - M421 HW 3 Due Friday Oct 5 From Wade Section 8.3 8.4...

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M421 HW 3 Due Friday Oct. 5 From Wade Section Page Number Problems 8.3 248 3, 5 8.4 254 9ab, 10a 9.1 262 4, 7, 8 Non-book Exercises 1) Suppose that A B R n . Prove that A B and A B . For the following exercises, the l p norm is defined on R n by bardbl vectorx bardbl p parenleftBigg n summationdisplay k =1 | x ( k ) | p parenrightBigg 1 p . 2) Prove for n = 2 that 1 2 bardbl vectorx bardbl 1 ≤ bardbl vectorx bardbl 2 . 3) Honors Option Problem. For this problem let p,q > 1 satisfy 1 p + 1 q = 1 . Such p and q are called conjugate exponents. (a) Prove Young’s inequality: | ab | ≤ | a | p p + | b | q q , for all conjugate exponents p,q and all a,b R . Hint: Compare the area of the box bounded by the lines x = 0 , y = 0 , x = a , and y = b to the area between the x axis and the curve y = x p 1 and between the y axis and the curve x = y q 1 . Use the fact that p,q conjugate exponents implies ( p 1)(

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