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
Nonbook 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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 Fall '07
 PROMISLOW
 Norm, Lp space, conjugate exponents

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