Is the Gram-Schmidt Orthogonalization process for functions the same as it is for vectors?

Question

I was not able to find resources online and was wondering so since it would greatly help me with my work if I can directly apply what I have learned from linear algebra.

Answer 1

Gram-Schmidt Orthonormalizing process can be applied to any (at most countable) linearly independent vectors in any inner product spaces. You can find the information at Wikipedia for finite vectors.

To apply the process for functions, we must prove that the functions we consider are in some inner product spaces (so that we can apply Gram-Schmidt through its inner product). Most commonly used inner product space for functions is $L^2$ space, which is the set of square-integrable functions (see this article in Wikipedia).

By the way, you can learn tools described above such as general inner product in functional analysis (as the generalization of linear algebra and topological theory for Euclid spaces). If you are not famillier with the notation in the article of above links, you can start with the standard textbook of functional analysis.