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 $\mathcal{N}(K)$.

For a bounded linear operator $K:U\to V$ between Hilbert spaces, a least-squares solution for $f$ exists exactly when $P_{\overline{\mathcal{R}(K)}}f\in \mathcal{R}(K)$, equivalently $f\in \mathcal{R}(K)\oplus \mathcal{R}(K{)}^{\perp }$. Indeed the orthogonal projection onto the closure uniquely minimizes distance to that closed space, and it can be realized by $Ku$ exactly under this condition. Thus all data admit least-squares solutions exactly when the range is closed.

The orthogonal projection of a prescribed point $u_0$ onto the affine subspace of least-squares solutions is unique. It retains $u_0$'s null space component and chooses the determined component from the Moore–Penrose inverse of an operator.