# Fredholm solvability

Let $A$ be a real $(n\times n)$-matrix and $b\in R^n$ a vector.

The Fredholm alternative in $R^n$ states that the equation $Ax=b$ has a solution if and only if $b^T v =0$ for every vector $v\in R^n$ satisfying $A^T v =0$.

This alternative has many applications, e.g. 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

$$(\ker T^*)^\perp = \{y\in F:(y,y^*)=0\;\forall y^* \in \ker T^*\}$$

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

The phrase "Fredholm solvability" refers to results and techniques for solving differential and integral equations via the Fredholm alternative and, more generally, the Fredholm-type properties of the operator involved.

How to Cite This Entry:
Fredholm solvability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_solvability&oldid=39569
This article was adapted from an original article by G. IsacThemistocles M. Rassias (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article