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