Fredholm solvability

{{no footnotes|date=October 2016}}

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 {{math|A}} be a real {{math|n × n}}-matrix and $b\backslash in\backslash mathbb\; R^n$ a vector.

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

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

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