# Fredholm solvability

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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.

How to Cite This Entry:
Fredholm solvability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_solvability&oldid=14350
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