= 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.
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