Let be a real -matrix and a vector.
The Fredholm alternative in states that the equation has a solution if and only if for every vector satisfying .
This alternative has many applications, e.g. in bifurcation theory. It can be generalized to abstract spaces. So, let and be Banach spaces (cf. Banach space) and let be a continuous linear operator. Let , respectively , denote the topological dual of , respectively , and let denote the adjoint of (cf. also Duality; Adjoint operator). Define
An equation is said to be normally solvable (in the sense of F. Hausdorff) if it has a solution whenever (cf. also Normal solvability). A classical result states that is normally solvable if and only if is closed in .
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, Fredholm-type properties of the operator involved.
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|[a3]||A.T. Prilepko, D.G. Orlovsky, I.A. Vasin, "Methods for solving inverse problems in mathematical physics" , M. Dekker (2000)|
|[a4]||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)|
Fredholm solvability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_solvability&oldid=14350