Fredholm theorem

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.

The Fredholm alternative is one of the Fredholm theorems.

Fredholm's theorem in linear algebra is as follows: if M is a matrix, then the orthogonal complement of the row space of M is the null space of M:

Similarly, the orthogonal complement of the column space of M is the null space of the adjoint:

Fredholm's theorem for integral equations is expressed as follows. Let ${\displaystyle K(x,y)}$ be an integral kernel, and consider the homogeneous equations

and its complex adjoint

Here, ${\displaystyle {\overline {\lambda }}}$ denotes the complex conjugate of the complex number ${\displaystyle \lambda }$, and similarly for ${\displaystyle {\overline {K(x,y)}}}$. Then, Fredholm's theorem is that, for any fixed value of ${\displaystyle \lambda }$, these equations have either the trivial solution ${\displaystyle \psi (x)=\phi (x)=0}$ or have the same number of linearly independent solutions ${\displaystyle \phi _{1}(x),\cdots ,\phi _{n}(x)}$, ${\displaystyle \psi _{1}(y),\cdots ,\psi _{n}(y)}$.

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