MATHEMATICS 116, FALL 2007
CONVEXITY AND OPTIMIZATION WITH APPLICATIONS
Assignment #4
Last modified: October 12, 2007
Due Thursday, Oct. 18 at class
This is short enough that you might have time left over to watch the baseball
playoffs.
1.
(a) Show that the function
f
(
x
) =
1
x
is continuous on (0,1], but not uni
formly continuous.
(b) Now consider any continuous functional
f
:
X
→
R
whose domain
includes a compact subset
S
of a normed vector space
X
. Show that
if
f
is continuous on
S
, it is uniformly continuous. Use no property
of compactness but the definition (Luenberger, p. 40). The example
in part (a) used an interval that was not compact. This is a famous
(named) theorem, so if you get stuck there is plenty of help available
online!
2.
(a) After consulting at least two sources, give a careful statement of the
“wellknown Weierstrass approximation theorem” to which Luenberger
alludes on page 42. Does the function being approximated have to be
uniformly continuous, or just continuous, or does it not matter?
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 Fall '09
 Math, Metric space, Compact space, normed vector space, Banach space, Luenberger

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