In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space
X
by first defining a linear transformation
T
on a dense subset of
X
and then extending
T
to the whole space via the theorem below. The resulting extension remains linear and bounded (thus continuous).
This procedure is known as continuous linear extension.
Every bounded linear transformation
T
from a normed vector space
X
to a complete, normed vector space
Y
can be uniquely extended to a bounded linear transformation
T
~
from the completion of
X
to
Y
. In addition, the operator norm of
T
is
c
iff the norm of
T
~
is
c
.
This theorem is sometimes called the B L T theorem, where B L T stands for bounded linear transformation.
Consider, for instance, the definition of the Riemann integral. A step function on a closed interval
[
a
,
b
]
is a function of the form:
f
≡
r
1
1
[
a
,
x
1
)
+
r
2
1
[
x
1
,
x
2
)
+
⋯
+
r
n
1
[
x
n
−
1
,
b
]
where
r
1
,
…
,
r
n
are real numbers,
a
=
x
0
<
x
1
<
…
<
x
n
−
1
<
x
n
=
b
, and
1
S
denotes the indicator function of the set
S
. The space of all step functions on
[
a
,
b
]
, normed by the
L
∞
norm (see Lp space), is a normed vector space which we denote by
S
. Define the integral of a step function by:
I
(
∑
i
=
1
n
r
i
1
[
x
i
−
1
,
x
i
)
)
=
∑
i
=
1
n
r
i
(
x
i
−
x
i
−
1
)
.
I
as a function is a bounded linear transformation from
S
into
R
.
Let
P
C
denote the space of bounded, piecewise continuous functions on
[
a
,
b
]
that are continuous from the right, along with the
L
∞
norm. The space
S
is dense in
P
C
, so we can apply the B.L.T. theorem to extend the linear transformation
I
to a bounded linear transformation
I
~
from
P
C
to
R
. This defines the Riemann integral of all functions in
P
C
; for every
f
∈
P
C
,
∫
a
b
f
(
x
)
d
x
=
I
~
(
f
)
.
The above theorem can be used to extend a bounded linear transformation
T
:
S
→
Y
to a bounded linear transformation from
S
¯
=
X
to
Y
, if
S
is dense in
X
. If
S
is not dense in
X
, then the Hahn–Banach theorem may sometimes be used to show that an extension exists. However, the extension may not be unique.
