Fredholm solvability

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 $$b\in\mathbb R^n$$ a vector.

The Fredholm alternative in $$\mathbb R^n$$ states that the equation $$Ax=b$$ has a solution if and only if $$b^T v =0$$ for every vector $$v\in\mathbb R^n$$ 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 $$T:E\rightarrow F$$ 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


 * $$(\ker T^*)^\perp = \{y\in F:(y,y^*)=0 \text{ for every } y^* \in \ker T^*\}$$

An equation $$Tx=y$$ is said to be normally solvable (in the sense of F. Hausdorff) if it has a solution whenever $$y \in (\ker T^*)^\perp$$. A classical result states that $$Tx=y$$ 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.