Fredholm solvability

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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.


[a1] F. Hausdorff, "Zur Theorie der linearen metrischen Räume" J. Reine Angew. Math. , 167 (1932) pp. 265
[a2] V.A. Kozlov, V.G. Maz'ya, J. Rossmann, "Elliptic boundary value problems in domains with point singularities" , Amer. Math. Soc. (1997)
[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)
How to Cite This Entry:
Fredholm solvability. Encyclopedia of Mathematics. URL:
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