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.
