ECE 580 / Math 587
FALL 2009
Correspondence
# 16
April 12, 2011
ASSIGNMENT 6
Reading Assignment:
Text: Chapter 5 (sects 5.105.13) and Chapter 6
Recommended Reading:
Curtain & Pritchard: pp. 6567, and Chapter 12;
Balakrishnan: pp. 6280, and Chapter 2.
Advance Reading:
Text: Chapter 7
Problems
(to be handed in):
Due Date:
Thursday, April 21.
41.
Let
X
be a Hilbert space, and
{
x
n
}
be a sequence in
X
, converging weakly to
x
o
∈
X
.
i)
Show
that the convergence is also in the strong sense (that is in norm) if further the
sequence of real numbers
{∥
x
n
∥}
converges to the norm of
x
o
,
∥
x
o
∥
.
ii)
Again for the original problem,
show
that one can find a subsequence
{
x
n
k
}
such that
the sequence of arithmetic means
y
m
=
1
m
m
∑
k
=1
x
n
k
,
m
= 1
,
2
, . . .
converges strongly to
x
0
(that is,
∥
y
m
−
x
o
∥ →
0).
42.
Let
X
be a real normed linear space, and
X
*
be its dual.
i)
Show
that a linear functional
f
on
X
is weakly continuous if and only if it is of the form
f
(
x
) =
< x, x
*
>
, for some
x
*
∈
X
*
.
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 Spring '08
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 Hilbert space, Banach, linear bounded operator

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