MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #8
Last modified: November 14, 2007
Due Tuesday, Nov. 20 at class.
1. Luenberger, Problem 5.14.1.
2. Luenberger, Problem 5.14.3. Explain why this result shows that the dual
of
l
∞
cannot be
l
1
.
Hint: Choose a countable dense subset of the unit sphere in
X
*
: the ele
ments of
X
*
for which

x
*

= 1. For each
x
*
n
in this subset, choose a vector
x
n
for which

< x
n

x
*
n
>

is close to 1 (say bigger than
1
2
). Now form the
closure of the space
Y
generated by all these
x
n
. Prove (easily) that this
space is separable. If it also equals
X
, you are done. So assume that
X
contains an element
x
0
that is not in
Y
, and show that this assumption
leads to a contradiction.
3. Luenberger, Problem 5.14.2.
This is challenging.
Both dual spaces are
isomorphic to
l
1
, but if the isomorphism is different they are “not identical.”
For
c
0
you can use an approach that parallels the one for
l
p
and
l
1
. It is
easy to show, using the Hoelder inequality, that any element of
l
1
defines
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 Fall '09
 Math, Linear Algebra, Vector Space, 2W, Dual space, countable dense subset, best approximates

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