ECE 580 / Math 587
SPRING 2011
Correspondence
# 14
March 30, 2011
ASSIGNMENT 5
Reading Assignment:
Text: Chapter 5, Sections 19
Advance Reading:
Text: Chapter 5, Sections 1013.
Recommended Reading:
Curtain & Pritchard: pp. 6567, and Chapter 12;
Balakrishnan: pp. 6280, and Chapter 2.
Problems
(to be handed in):
Due Date:
Tuesday, April 12
.
34.
Let
X
=
C
[0
,
1] with the standard maximum norm, and
f
∈
X
*
be defined by
f
(
x
) =
x
(1
/
4) + 3
x
(1
/
2) + 2
∫
1
0
tx
(
t
)
dt
i)
Compute
∥
f
∥
, the norm of
f
.
ii)
Obtain
a function of bounded variation on [0
,
1], say
v
(
·
), such that
v
(0) = 0 and
f
(
x
) =
∫
1
0
x
(
t
)
dv
(
t
)
35.
Let
g
1
, g
2
, . . . , g
n
be linearly independent linear functionals on a vector space
X
. Let
f
be
another linear functional on
X
such that for every
x
∈
X
satisfying
g
i
(
x
) = 0
, i
= 1
,
2
, . . . , n
,
we have
f
(
x
) = 0.
Show
that there exist constants
λ
1
, λ
2
, . . . , λ
n
such that
f
=
n
∑
i
=1
λ
i
g
i
.
Hint :
Use the HahnBanach Theorem (extension form; Correspondence 13).
36.
Let
c
0
be the space of all infinite sequences of real numbers converging to
zero
, endowed with
the
max
norm. Let
x
=
{
ξ
i
}
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Linear Algebra, Vector Space, Topological vector space, unit norm

Click to edit the document details