Generalized Orthogonalization (of which the Gram–Schmidt process is merely a minor consequence)

This paper establishes a generalized framework for orthogonaliza
tion, in which the classical Gram–Schmidt process appears only as a
particular and relatively minor consequence. We rigorously reconstruct
the fundamental definitions, lemmas, and computations. The paper
emphasizes both the algebraic structure and the inductive methodol
ogy that govern the construction of orthogonal systems. Our results
highlight the generality of this approach and provide a foundation for
further developments in linear algebra and its applications