Chapter 5
Topological Vector Spaces
In this chapter
V
is a real or complex vector space.
5.1 Topological Vector Spaces
A complex vector space
V
equipped with a topology is a
broadsense topo
logical vector space
if the mappings
V
×
V
→
V
: (
x,y
)
7→
x
+
y
C
×
V
→
V
: (
λ,x
)
7→
λx
are continuous. Observe that then, for each
x
∈
V
, the translation map
τ
x
:
V
→
V
:
y
7→
y
+
x
is continuous. Since
τ

1
x
=
τ

x
, it follows that
τ
x
is a homeomorphism.
The simple but important consequence of this is that
V
“looks the same
everywhere”, i.e.
if
a,b
∈
V
then there is a homeomorphism
, speciﬁcally
τ
b

a
:
V
→
V
,
which maps
a
to
b
. In particular, every neighborhood of
x
∈
V
is a translate of a neighborhood of 0, i.e. of the form
x
+
U
for some
neighborhood
U
of 0. For this reason, we shall prove most of our results in
a neighborhood of 0.
By a
topological vector space
we shall mean a broadsense topological
vector space which is
Hausdorﬀ
, i.e. distinct points of disjoint neighborhoods.
Lemma 1
Let
V
be a broadsense topological vector space, and
W
an open
set with
0
∈
W
. Then there is an open set
U
with
0
∈
U
,
U
=

U
, and
U
+
U
⊂
W
1
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CHAPTER 5. TOPOLOGICAL VECTOR SPACES
Proof
. Since
V
×
V
→
V
: (
x,y
)
7→
x
+
y
is continuous at 0, and
W
is a
neighborhood of 0, there is a neighborhood
U
1
of 0 such that
U
1
+
U
1
⊂
W
To get symmetry take
U
=
U
1
∩
(

U
1
)
Note that continuity of multiplication by scalars implies that
x
7→ 
x
is a
homeomorphism and so

U
1
is open when
U
1
is open.
QED
Here is a simple but useful observation: if
A
is any subset of the broad
sense topological vector space
V
and
U
an open subset of
V
then
the set
A
+
U
=
{
a
+
x
:
a
∈
A,x
∈
U
}
is open.
The reason is that
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 Spring '02
 Sengupta
 Logic, Topology, Vector Space, Metric space, Open set, Topological space, 5 w

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