ECE 580 / Math 587
SPRING 2011
Professor Tamer Ba¸
sar
May 5, 2011
FINAL EXAM
DUE DATE & TIME :
Monday, May 9, 2011; 9:00 am.
If I am not around, leave with my secretary next door to my office at CSL, or
slide under my door.
Answer the four questions below, starting each one on a separate sheet.
Problem 1
Let
C
[
−
1
,
1] denote the normed vector space of continuous functions on [
−
1
,
1], equipped with
the standard maximum norm. We wish to find a linear functional
f
on
X
=
C
[
−
1
,
1], with
minimum norm, that satisfies the side conditions:
f
(
x
1
) = 1
,
f
(
x
2
) = 2
where
x
1
(
t
) = 2
t
2
−
t ,
x
2
(
t
) =
−
t
2
+
t
+ 1
i)
Is this a feasible problem?
Justify your answer.
ii)
If your answer to part
(i)
is in the affirmative, obtain the solution.
iii)
Repeat
(i)
and
(ii)
above with
X
=
L
2
[
−
1
,
1], instead of
C
[
−
1
,
1], where
L
2
[
−
1
,
1] is the
standard Hilbert space of squareintegrable functions on [
−
1
,
1].
iv)
Would the result in
(iii)
be any different if
X
is the normed vector space of continuous
functions on [
−
1
,
1] with the
L
2
norm?
v)
Let
x
*
be the solution you obtained in part
(ii)
. Describe the set of all
x
∈
C
[
−
1
,
1] that
are orthogonal to
x
*
, that is
< x, x
*
>
= 0.
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 Spring '08
 Staff
 Derivative, Vector Space, Banach space, XO, standard Hilbert space

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