In the mathematics of Banach spaces, the method of continuity provides sufficient conditions for deducing the invertibility of one bounded linear operator from that of another, related operator.
Let B be a Banach space, V a normed vector space, and
(
L
t
)
t
∈
[
0
,
1
]
a norm continuous family of bounded linear operators from B into V. Assume that there exists a constant C such that for every
t
∈
[
0
,
1
]
and every
x
∈
B
|
|
x
|
|
B
≤
C
|
|
L
t
(
x
)
|
|
V
.
Then
L
0
is surjective if and only if
L
1
is surjective as well.
The method of continuity is used in conjunction with a priori estimates to prove the existence of suitably regular solutions to elliptic partial differential equations.
We assume that
L
0
is surjective and show that
L
1
is surjective as well.
Subdividing the interval [0,1] we may assume that
|
|
L
0
−
L
1
|
|
≤
1
/
(
3
C
)
. Furthermore, the surjectivity of
L
0
implies that V is isomorphic to B and thus a Banach space. The hypothesis implies that
L
1
(
B
)
⊆
V
is a closed subspace.
Assume that
L
1
(
B
)
⊆
V
is a proper subspace. The Hahn–Banach theorem shows that there exists a
y
∈
V
such that
|
|
y
|
|
V
≤
1
and
d
i
s
t
(
y
,
L
1
(
B
)
)
>
2
/
3
. Now
y
=
L
0
(
x
)
for some
x
∈
B
and
|
|
x
|
|
B
≤
C
|
|
y
|
|
V
by the hypothesis. Therefore
|
|
y
−
L
1
(
x
)
|
|
V
=
|
|
(
L
0
−
L
1
)
(
x
)
|
|
V
≤
|
|
L
0
−
L
1
|
|
|
|
x
|
|
B
≤
1
/
3
,
which is a contradiction since
L
1
(
x
)
∈
L
1
(
B
)
.
