 Affine solution space of a linear equation

For a linear map $L:V\to W$ and one solution $L{u}_0=f$, 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.