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

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)(