1Normed vector spaces
II Linear Analysis
1.1 Bounded linear maps
With vector spaces, we studied linear maps. These are maps that respect the
linear structure of a vector space. With normed vector spaces, the right kind of
maps to study is the bounded linear maps.
Definition
(Bounded linear map)
. T
:
X → Y
is a bounded linear map if there
is a constant
C >
0 such that
kT xk
Y
≤ Ckxk
X
for all
x ∈ X
. We write
B
(
X, Y
)
for the set of bounded linear maps from X to Y .
This is equivalent to saying
T
(
B
X
(1))
⊆ B
Y
(
C
) for some
C >
0. This also
equivalent to saying that
T
(
B
) is bounded for every bounded subset
B
of
X
.
Note that this final characterization is also valid when we just have a topological
vector space.
How does boundedness relate to the topological structure of the vector spaces?
It turns out that boundedness is the same as continuity, which is another reason
why we like bounded linear maps.
Proposition.
Let
X
,
Y
be normed vector spaces,
T
:
X → Y
a linear map.
Then the following are equivalent:
(i) T is continuous.
(ii) T is continuous at 0.
(iii) T is bounded.
Proof. (i) ⇒ (ii) is obvious.
(ii)
⇒
(iii): Consider
B
Y
(1)
⊆ Y
, the unit open ball. Since
T
is continuous
at 0,
T
−1
(
B
Y
(1))
⊆ X
is open. Hence there exists
ε >
0 such that
B
X
(
ε
)
⊆
T
−1
(B
Y
(1)). So T (B
x
(ε)) ⊆ B
Y
(1). So T (B
X
(1)) ⊆ B
Y
1
ε
. So T is bounded.
(iii)
⇒
(i): Let
ε >
0. Then
kT x
1
−T x
2
k
Y
=
kT
(
x
1
−x
2
)
k
Y
≤ Ckx
1
−x
2
k
X
.
This is less than ε if kx
1
− x
2
k < C
−1
ε. So done.
Using the obvious operations,
B
(
X, Y
) can be made a vector space. What
about a norm?
Definition
(Norm on
B
(
X, Y
))
.
Let
T
:
X → Y
be a bounded linear map.
Define kT k
B(X,Y )
by
kT k
B(X,Y )
= sup
kxk≤1
kT xk
Y
.
Alternatively, this is the minimum
C
such that
kT xk
Y
≤ Ckxk
X
for all
x
.
In particular, we have
kT xk
Y
≤ kT k
B(X,Y )
kxk
X
.