View source for 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 (cf. [[Banach space|Banach space]]) and let $T:E\rightarrow F$ be a continuous [[Linear operator|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|Duality]]; [[Adjoint operator|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|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.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F. Hausdorff, "Zur Theorie der linearen metrischen Räume" ''J. Reine Angew. Math.'' , '''167''' (1932) pp. 265 {{MR|}} {{ZBL|0003.33104}} {{ZBL|58.1113.05}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.A. Kozlov, V.G. Maz'ya, J. Rossmann, "Elliptic boundary value problems in domains with point singularities" , Amer. Math. Soc. (1997) {{MR|1469972}} {{ZBL|0947.35004}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.T. Prilepko, D.G. Orlovsky, I.A. Vasin, "Methods for solving inverse problems in mathematical physics" , M. Dekker (2000) {{MR|1748236}} {{ZBL|0947.35173}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> D.G. Orlovskij, "The Fredholm solvability of inverse problems for abstract differential equations" A.N. Tikhonov (ed.) et al. (ed.) , ''Ill-Posed Problems in the Natural Sciences'' , VSP (1992) {{MR|}} {{ZBL|0789.35178}} </TD></TR></table>