---
tag: 242TFD41KZ
kind: concept
name: Fredholm theory
msc: [45B05, 47A53, 47B07]
sources:
  - url: https://en.wikipedia.org/wiki/Fredholm_theory
same_as:
  wikidata: Q4455098
  wikipedia: Fredholm theory
---

## Summary

Fredholm theory is the theory of the [[Fredholm equation of the second kind|DEYYNVCV9J]] $\varphi - \lambda T_K \varphi = f$.
In its classical form it builds the solution from the [[Fredholm determinant|PPX0AK81QV]] and the [[Fredholm minors|HP5VJ9EBST]], entire functions of $\lambda$.
It proves [[Fredholm's theorem|XQ8DYKW4SH]]: the equation behaves like a square linear system.
Its abstract form is the [[spectral theory of compact operators|JEPDM5B5FC]] on Banach spaces, and it leads to the notion of a [[Fredholm operator|8HWGKWD6ZM]].
It was one of the sources of functional analysis.

## Explanations

### L1 human

A square system of linear equations either has exactly one solution for each right-hand side, or its homogeneous version has a nonzero solution.
Fredholm theory shows that integral equations of the second kind behave in the same way, although they involve infinitely many unknowns, one value of $\varphi$ for each point.
The trick is to replace the integral by a finite sum, solve the resulting linear system with determinants, and let the number of points grow.
For a kernel such as $K(x, y) = xy$, the integral equation reduces exactly to one linear equation in one unknown, and the theory can be checked by hand.