MAT 578 HW 1 Due Tuesday, 8/28/01
1.
Let
X
be a TVS and
Y
X
a nite dimensional subspace. Provethat
Y
is a closed
subset of
X
.
2.
Let
U
be a subset of a vector space
X
. Suppose that
U
is an absorbing set, and that
for all
x
2
U
,
sx
2
U
whenever 0
s
1. (Recall that
U
is
absorbing
if for every
x
2
X
,
there exists
t>
0such that
x
2
tU
.) Let
p
U
be the gauge of
U
:
p
U
(
x
) = inf
t>
0
x
2
tU
:
Prove the following statements.
(i) (
8
x
2
X
)(
8
t
0)
p
U
(
tx
)=
tp
U
(
x
)
.
(ii)
x
2
X
p
U
(
x
)
<
1
U
x
2
X
p
U
(
x
)
1
.
(iii) If
U
is balanced, then (
8
x
2
X
)(
8
scalar
)
p
U
(
x
)=
j
j
p
U
(
x
)
.
(iv) If
U
is convex, then (
8
x; y
2
X
)
p
U
(
x
+
y
)
p
U
(
x
)+
p
U
(
y
)
.
3.
Let
X
be a vector space, and let
E
be a nondegenerate family of seminorms on
X
.
Prove that scalar multiplication is continuous for the top ology on
X
determined by
E
.