An algorithmic procedure that transforms a finite linearly independent set in an inner product space into an orthonormal set spanning the same subspace by successive orthogonal projection and normalization.

Principle

At each step subtract the projection of the current vector onto the span of previously constructed orthonormal vectors to enforce orthogonality, then scale to unit norm to enforce normalization.

Demonstration

Starting with two independent vectors in R^2, subtracting the projection of the second onto the first and normalizing yields an orthonormal pair aligned with the original span.

Misapplication

Applying the classical algorithm in finite-precision arithmetic without reorthogonalization on nearly linearly dependent inputs leads to loss of orthogonality and numerical instability.

Consequence

Produces an orthonormal basis for the span, enabling stable coordinate expansion, projection computations, and construction of orthogonal factorizations of matrices when implemented with numerical care.

Reversal

Failure of orthogonality signals linear dependence or catastrophic cancellation; reverting to rank-revealing factorizations or stabilized variants recovers robust orthonormal bases.

Boundary

Requires an inner product (or preinner product) and nonzero initial vectors; infinite or continuous sets require functional-analytic generalizations and are not covered by the finite-step algorithm.

Semantic Tension

Related method: matrix factorization approaches produce equivalent orthogonal bases via different numerical strategies; the Gram–Schmidt formula is constructive but may be less stable than alternative factorizations unless modified.

Synthesis

A constructive orthogonalization algorithm that iteratively removes components along already constructed directions and normalizes, yielding an orthonormal basis of the original finite-dimensional subspace when properly stabilized.