 Least-squares solution of a linear inverse problem

A least-squares solution minimizes $\Vert Ku-f\Vert$ over the source Hilbert space. It exists exactly when $f\in \mathcal{R}(K)\oplus \mathcal{R}(K{)}^{\perp }$, and then the solutions form a closed affine subspace with direction $\mathcal{N}(K)$.