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ECE 580, Optimization by Vector Space Methods
Assignment # 3
Issued: February 6
Due: February 20, 2008
Reading Assignment:
Luenberger
, complete Chapter 3, and begin Chapter 5.
Problems:
10 Before completing this exercise take a look at the Riesz Lemma
See for example
http://en.wikipedia.org/wiki/Riesz’s_lemma
(i) Let
Y
be a proper closed subset of the normed linear space
X
(not necessarily a
innerproduct space). Then, given
ε >
0 there exists
x
∈
X
satisfying
b
x
b
= 1
and
d
(
x,Y
)
≥
1
−
ε
. Furthermore, if
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Unformatted text preview: Y is ±nitedimensional one can choose x so that d ( x,Y ) = 1. What if X is a Hilbert space ? (ii) Let X be an in±nite dimensional normed linear space. Show that there is a sequence { x n } ⊂ X satisfying b x b = 1 and b x n − x m b ≥ 1 for each n,m . (iii) Conclude that the closed unit ball is not compact if X is in±nitedimensional. 11 Luenberger Prob. 3.7 12 Luenberger Prob. 3.12 13 Luenberger Prob. 3.21 14 Luenberger Prob. 3.22...
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This note was uploaded on 10/11/2011 for the course ECE 580 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Staff

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