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The 19 pages that link to Orthogonal projection, each with the reason it gives.

• Hilbert spaceRelated: Completeness guarantees that every closed subspace has a well-defined nearest-point projection.

• Least squaresRelated: The fitted values are the projection of observations onto the model’s column space.

• Cauchy–Schwarz inequalityRelated: Projecting one vector onto the direction of another gives a geometric proof of the bound.

• Inner productRelated: Inner products characterize the residual as orthogonal to the projected subspace.

• Orthonormal basisRelated: Finite-dimensional projections onto basis spans converge to the represented vector.

• Inner product spaceRelated: It uses inner products to split vectors into parallel and perpendicular components.

• Orthogonal matrixCompared with: Unlike an orthogonal matrix, a projection usually shortens vectors and is not invertible.

• Dihedral angleRelated: Projecting perpendicular to the common line reduces the plane angle to an ordinary planar angle.

• Adjoint operatorRelated: Its adjoint equals itself, illustrating the symmetry encoded by the defining identity.

• Parallelogram lawRelated: The law supports distance decompositions used to characterize best approximations by projections.

• Column spaceRelated: Least-squares fitting projects the target vector onto the matrix’s column space.

• Bessel's inequalityRelated: Projecting onto finitely many orthonormal vectors yields the norm bound directly.

• Sum of squaresRelated: Least-squares fitted values are projections, making residuals orthogonal to the model space.

• Skew linesRelated: Projecting a skew pair can make their spatial separation easier to analyze.

• Euclidean vectorRelated: It decomposes a vector into parallel and perpendicular components.

• Law of total varianceRelated: In square-integrable settings, the decomposition follows from orthogonality of prediction and residual.

• Cochran's theoremRelated: Orthogonal projections generate the independent quadratic forms in the theorem’s standard decomposition.

• Frisch–Waugh–Lovell theoremBroader topic: Regressing out controls is precisely projection onto their column space.

• Riesz's lemmaCompared with: Riesz's lemma applies without an inner product or a nearest point in the subspace.