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Unformatted text preview: M421 HW 3 Due Friday Oct. 5
From Wade Section 8.3 8.4 9.1 Page Number 248 254 262 Problems 3, 5 9ab, 10a 4, 7, 8 Non-book Exercises
1) Suppose that A ⊂ B ⊂ Rn . Prove that A ⊂ B and A◦ ⊂ B ◦ . For the following exercises, the lp norm is deﬁned on Rn by
n x p ≡
k=1 |x(k)| p 1 p . 2) Prove for n = 2 that 1 √ 2 x 1 ≤ x 2. 3) Honors Option Problem. For this problem let p, q > 1 satisfy are called conjugate exponents. (a) Prove Young’s inequality: |ab| ≤ |a|p |b|q + , p q 1 p + 1 q = 1. Such p and 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 = xp−1 and between the y axis and the curve x = y q−1 . Use the fact that p, q conjugate exponents implies (p − 1)(q − 1) = 1, so the two curves are the same. (b) Use Young’s inequality to prove H¨lder’s inequality: o
n |x(k)y(k)| ≤ x
k=1 p y q, for all conjugate exponents p and q. 1 (c) Use H¨lder’s inequality to prove that o 1 n1/q x
1 ≤ x p, for all x ∈ Rn and all conjugate exponents p and q. (d) Use H¨lder’s inequality to prove the triangle inequality for the lp norm, p ≥ 1. That is o x+y for all x, y ∈ Rn .
p ≤ x p + y p, 2 ...
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This note was uploaded on 07/25/2008 for the course MATH 421 taught by Professor Promislow during the Fall '07 term at Michigan State University.
- Fall '07