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  "body": "\n## Summary\n\nLet an [[integral equation of the second kind|8KJCN49V64]] have kernel $K(x, s)$ and parameter $\\lambda$.\nIts adjoint is the equation of the same kind with the kernel $\\overline{K(s, x)}$ and the parameter $\\bar\\lambda$.\nIts integral operator is the [[adjoint|0R9NMSM1ZP]] of the original integral operator for the $L^2$ inner product.\nThe Fredholm alternative uses it: the original equation is solvable exactly when its right side is orthogonal to the solutions of the homogeneous adjoint equation.\nFor a real kernel the adjoint kernel is the transposed kernel $K(s, x)$.\n\n## Explanations\n\n### L1 human\n\nThe adjoint equation swaps the roles of the two variables of the kernel, like transposing a matrix.\nFor complex kernels one also takes complex conjugates, as in the conjugate transpose of a matrix.\nThe original equation and its adjoint have solvability properties that mirror each other.\nFor example, the adjoint of $\\varphi(x) = f(x) + \\lambda \\int_0^1 x \\, \\varphi(s) \\, ds$ is $\\psi(x) = g(x) + \\bar\\lambda \\int_0^1 s \\, \\psi(s) \\, ds$.\n",
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