MATH 7338
HOMEWORK #3
Due date: October 29, 2009
Work FIVE of the following problems and hand in your solutions. You may work together
with other people in the class, but you must each write up your solutions independently. A
subset of these will be selected for grading. Write LEGIBLY on the FRONT side of the page
only, and STAPLE your pages together.
1.
Let
F
(
R
) be the vector space containing all functions
f
:
R
→
C
.
For each
x
∈
R
,
define a seminorm on
F
(
R
) by
ρ
x
(
f
) =

f
(
x
)

. Then convergence with respect to the family
of seminorms
{
ρ
x
}
x
∈
R
corresponds to pointwise convergence of functions.
Show that this
topology is not normable, i.e., Show that there is no norm on
F
(
R
) that defines the same
convergence criterion.
Hint: Find
f
n
such that
c
n
f
n
→
0 pointwise for every choice of scalars
c
n
.
2.
Let
{
ρ
α
}
α
∈
J
be a family of seminorms on a vector space
X.
Show that the induced
topology on
X
is Hausdorff if and only if
ρ
α
(
x
) = 0 for all
α
∈
J
⇐⇒
x
= 0
.
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.
 Fall '09
 HEIL
 Math, Topology, µ, Hilbert space, General topology, C0, Weak topology

Click to edit the document details