= Affine solution space of a linear equation
{title2=$u_0+\ker L$}
For a <linear map> $L:V\to W$ and one solution $Lu_0=f$, all solutions are $u_0+\ker L$. This follows because $L(u-u_0)=0$, and conversely adding any kernel element preserves the equation. If $U\subseteq V$ is a <vector subspace> representing a desired regularity class and $u_0\in U$, then every solution to this one inhomogeneous equation lies in $U$ exactly when $\ker L\subseteq U$. For distributional differential equations, take $U$ to be the distributions represented by <smooth functions>.
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