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  "body": "\n## Summary\n\nFredholm theory is the theory of the [[Fredholm equation of the second kind|DEYYNVCV9J]] $\\varphi - \\lambda T_K \\varphi = f$.\nIn its classical form it builds the solution from the [[Fredholm determinant|PPX0AK81QV]] and the [[Fredholm minors|HP5VJ9EBST]], entire functions of $\\lambda$.\nIt proves [[Fredholm's theorem|XQ8DYKW4SH]]: the equation behaves like a square linear system.\nIts abstract form is the [[spectral theory of compact operators|JEPDM5B5FC]] on Banach spaces, and it leads to the notion of a [[Fredholm operator|8HWGKWD6ZM]].\nIt was one of the sources of functional analysis.\n\n## Explanations\n\n### L1 human\n\nA square system of linear equations either has exactly one solution for each right-hand side, or its homogeneous version has a nonzero solution.\nFredholm 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.\nThe trick is to replace the integral by a finite sum, solve the resulting linear system with determinants, and let the number of points grow.\nFor 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.\n",
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