In mathematics, Fredholm solvability encompasses results and techniques for solving differential and integral equations via the Fredholm alternative and, more generally, the Fredholm-type properties of the operator involved. The concept is named after Erik Ivar Fredholm.
Let A be a real n × n-matrix and
b
∈
R
n
a vector.
The Fredholm alternative in
R
n
states that the equation
A
x
=
b
has a solution if and only if
b
T
v
=
0
for every vector
v
∈
R
n
satisfying
A
T
v
=
0
. This alternative has many applications, for example, in bifurcation theory. It can be generalized to abstract spaces. So, let
E
and
F
be Banach spaces and let
T
:
E
→
F
be a continuous linear operator. Let
E
∗
, respectively
F
∗
, denote the topological dual of
E
, respectively
F
, and let
T
∗
denote the adjoint of
T
(cf. also Duality; Adjoint operator). Define
(
ker
T
∗
)
⊥
=
{
y
∈
F
:
(
y
,
y
∗
)
=
0
for every
y
∗
∈
ker
T
∗
}
An equation
T
x
=
y
is said to be normally solvable (in the sense of F. Hausdorff) if it has a solution whenever
y
∈
(
ker
T
∗
)
⊥
. A classical result states that
T
x
=
y
is normally solvable if and only if
T
(
E
)
is closed in
F
.
In non-linear analysis, this latter result is used as definition of normal solvability for non-linear operators.
