= Solvability condition
A solvability condition is a compatibility requirement on data for an equation to admit a solution in a specified function space or with specified boundary behavior. For a bounded <linear operator> between <Hilbert spaces>, the equation $Lu=f$ requires $f$ to be orthogonal to the kernel of the adjoint. A closed range makes that requirement sufficient; the <Fredholm alternative for a compact operator> applies this principle to $I-K$ with compact $K$. in an asymptotic inner problem, exclusion of a growing homogeneous mode can impose an analogous weighted-integral condition.
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