Least-squares existence criterion

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.