This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 Nonbook 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 ≤ ap bq + , 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 ...
View
Full
Document
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
 PROMISLOW

Click to edit the document details