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In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra , a theorem of integral equations , or as a theorem on Fredholm operators .
In mathematics, Fredholm solvability encompasses results and techniques for solving differential and integral equations via the Fredholm alternative and, more generally, the Fredholm-type properties of the operator involved. The concept is named after Erik Ivar Fredholm. Let A be a real n × n-matrix and a vector
In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the spectral theory of Fredholm operators and Fredholm kernels on Hilbert space.
In mathematics, Fredholm's theorems are a set of celebrated results of Ivar Fredholm in the Fredholm theory of integral equations. There are several closely related theorems, which may be stated in terms of integral equations, in terms of linear algebra , or in terms of the Fredholm operator on Banach spaces .
A crucial property of compact operators is the Fredholm alternative, which asserts that the existence of solution of linear equations of the form ( λ K + I ) u = f {\displaystyle (\lambda K+I)u=f} (where K is a compact operator, f is a given function, and u is the unknown function to be solved for) behaves much like as in finite dimensions.
In mathematics, the analytic Fredholm theorem is a result concerning the existence of bounded inverses for a family of bounded linear operators on a Hilbert space. It is the basis of two classical and important theorems, the Fredholm alternative and the Hilbert–Schmidt theorem. The result is named after the Swedish mathematician Erik Ivar ...
In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian.
Erik Ivar Fredholm (7 April 1866 – 17 August 1927) was a Swedish mathematician whose work on integral equations and operator theory foreshadowed the theory of Hilbert spaces. Biography [ edit ]