= Fredholm solvability condition for a self-adjoint operator
{c}
For a <self-adjoint operator> $L$, solving $Lu=f$ requires $f$ to be <orthogonal> to every $v\in\ker L$, because $\langle v,Lu\rangle=\langle Lv,u\rangle=0$. When the range is closed, this condition is also sufficient: $(\operatorname{ran}L)^\perp=\ker L$ and closedness gives $\operatorname{ran}L=(\ker L)^\perp$. In finite dimensions the range is automatically closed. This <Fredholm solvability condition> determines <amplitude equations> by projecting perturbative forcing onto critical <eigenfunctions>.
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