Construction of linearly independent and orthogonal functions in Hilbert function spaces via Wronski determinants

Based on the Wronski determinant, we propose the construction of linearly independent and orthogonal functions in any Hilbert function space. The method requires only an initial function from the space of functions under consideration, that satisfies mild conditions, and emerges as a generalization of the Gram-Schmidt process. Two applications are considered, including solutions to ordinary differential equations and the construction of basis functions. We also present a conjecture that connects the latter two concepts, which leads to the introduction of the Wronski basis.