Editing Fredholm operator (section)

==Applications==
Any [[elliptic operator]] on a closed manifold can be extended to a Fredholm operator. The use of Fredholm operators in [[partial differential equation]]s is an abstract form of the [[parametrix]] method.

The [[Atiyah-Singer index theorem]] gives a topological characterization of the index of certain operators on manifolds.

The [[Atiyah-Jänich theorem]] identifies the [[Topological K-theory|K-theory]] ''K''(''X'') of a compact topological space ''X'' with the set of [[homotopy class]]es of continuous maps from ''X'' to the space of Fredholm operators ''H''&rarr;''H'', where ''H'' is the separable Hilbert space and the set of these operators carries the operator norm.